UNCERTAINTY-DRIVEN ADAPTIVE ESTIMATION WITH …

UNCERTAINTY-DRIVEN ADAPTIVE ESTIMATION WITHAPPLICATIONS IN ELECTRICAL POWER SYSTEMS

Jinghe Zhang

A dissertation submitted to the faculty of the University of North Carolina at Chapel Hill inpartial fulfillment of the requirements for the degree of Doctor of Philosophy in the

Department of Computer Science

Chapel Hill2013

Approved by:

Gregory F. Welch

Gary Bishop

Ning Zhou

Zhenyu Huang

Montek Singh

c⃝ 2013Jinghe Zhang

ALL RIGHTS RESERVED

ii

ABSTRACTJINGHE ZHANG: Uncertainty-Driven Adaptive Estimation with Applications in Electrical

Power Systems(Under the direction of Gregory F. Welch)

From electrical power systems to meteorology, large-scale state-space monitoring and

forecasting methods are fundamental and critical. Such problem domains pose challenges from

both computational and signal processing perspectives, as they typically comprise a large num-

ber of elements, and processes that are highly dynamic and complex (e.g., severe nonlinearity,

discontinuities, and uncertainties). This makes it especially challenging to achieve real-time

operations and control.

For decades, researchers have developed methods and technology to improve the accu-

racy and efficiency of such large-scale state-space estimation. Some have devoted their efforts

to hardware advances—developing advanced devices with higher data precision and update

frequency. I have focused on methods for enhancing and optimizing the state estimation per-

formance.

As uncertainties are inevitable in any state estimation process, uncertainty analysis can

provide valuable and informative guidance for on-line, predictive, or retroactive analysis. My

research focuses primarily on three areas:

1. Grid Sensor Placement. I present a method that combines off-line steady-state uncer-

tainty and topology analysis for optimal sensor placement throughout the grid network.

2. Filter Computation Adaptation. I present a method that utilizes on-line state uncertainty

analysis to choose the best measurement subsets from the available (large-scale) measurement

data. This allows systems to adapt to dynamically available computational resources.

3. Adaptive and Robust Estimation. I present a method with a novel on-line measurement

uncertainty analysis that can distinguish between suboptimal/incorrect system modeling and/or

iii

erroneous measurements, weighting the system model and measurements appropriately in real-

time as part of the normal estimation process.

We seek to bridge the disciplinary boundaries between Computer Science and Power Sys-

tems Engineering, by introducing methods that leverage both existing and new techniques.

While these methods are developed in the context of electrical power systems, they should

generalize to other large-scale scientific and engineering applications.

iv

Dedicated to

My Family

v

ACKNOWLEDGEMENTS

When I first stepped into Sitterson Hall, our computer science department at UNC Chapel

Hill, I did not know what to expect from my new graduate student life. And honestly, I did not

expect that all things can fall into place so smoothly for me, for which I am very thankful.

Since I started writing this acknowledgements section, I tried to list all the people who have

helped my graduate career, and then eventually I realized that would be mission-impossible.

Because there are so many people who have inspired, supported and helped me, in one way

or another. So as I write this acknowledgement to express my sincere gratitude, I apologize in

advance to anyone who has been left out, and hope that I can at least remember those who have

influenced me the most along the way. Many thanks to:

• Dr. Greg Welch, for giving me the opportunity to work as a research assistant, for being

my advisor and friend, and for being the most enthusiastic, insightful and knowledgeable

computer scientist that I have had the pleasure of working with;

• Dr. Gary Bishop for constantly inspiring me with his creative ideas, profound knowledge

and cheerful personality, and for passing on “rationalizing everything" ability to me;

• Dr. Montek Singh for agreeing to serve on my committee, and for his support and in-

sightful feedbacks;

• Dr. Zhenyu Huang for supporting me in all possible ways, and for hosting us during our

visits to the Pacific Northwest National Lab (PNNL);

• Dr. Ning Zhou for his thoughtful advice and help on my research progress, and for

providing helpful background information;

• Dr. Pengwei Du and Dr. Ruisheng Diao, for providing “real-time technique support"

whenever I have questions about power systems;

vi

• Janet Jones for helping me and looking out for me before her retirement, and for being a

friend indeed (and in need) ever after;

• Dawn Andres for making my travel and commuting so easy and worry-free;

• Tianren Wang and Rukun Fan for being my friends and teammates. We have been help-

ing, supporting and sharing ideas with each other unconditionally.

On a more personal note, I would like to thank:

• My family, for their love, patience and support. They are proud of me as much as I am

proud of them. Thanks for being my rock!

Financial support for this work came from:

• U.S. Department of Energy grant DE-SC0002271 “Advanced Kalman Filter for Real-

Time Responsiveness in Complex Systems," PIs Zhenyu Huang at PNNL and Greg

Welch at UNC. At DOE we acknowledge Sandy Landsberg, Program Manager for Ap-

plied Mathematics Research; Office of Advanced Scientific Computing Research; DOE

Office of Science;

• The Office of Naval Research award N00014-09-1-0813, “3D Display and Capture of

Humans for Live-Virtual Training." We acknowledge Dr. Roy Stripling, Program Man-

ager at ONR.

• The Office of Naval Research award N00014-12-1-0052, “3D Display and Capture of

Humans for Live-Virtual Training." We acknowledge Peter Squire, HPTE Deputy Thrust

Manager at ONR.

vii

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Motivations and Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Grid Sensor Placement . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Filter Computation Adaptation . . . . . . . . . . . . . . . . . . . . . . 4

1.1.3 Adaptive and Robust Estimation . . . . . . . . . . . . . . . . . . . . . 5

1.2 Thesis Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Grid Sensor Placement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Chapter Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Background and Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 State Space Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5 Steady State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.6 Traditional Power System State Estimation . . . . . . . . . . . . . . . . . . . . 17

viii

2.6.1 Weighted Least Squares Estimator . . . . . . . . . . . . . . . . . . . . 17

2.6.2 The Measurement Function and Measurement Jacobian . . . . . . . . . 19

2.7 The Phasor Measurement Units (PMUs) . . . . . . . . . . . . . . . . . . . . . 23

2.7.1 Observability Checking . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.8 My Method: PMU Placement for Power System Static State Estimation . . . . 25

2.8.1 PMU Placement Evaluation . . . . . . . . . . . . . . . . . . . . . . . . 25

2.8.2 Optimal PMU Placement . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.8.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.9 My Method: PMU Placement for Power System Dynamic State Estimation . . 41

2.9.1 PMU Placement Evaluation . . . . . . . . . . . . . . . . . . . . . . . . 41

2.9.2 Optimal PMU Placement . . . . . . . . . . . . . . . . . . . . . . . . . 46


3 Filter Computation Adaptation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52



3.4 The Kalman Filter Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4.1 The Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4.2 The Extended Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . 63

3.5 The SCAAT Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.6 My Method: Lower Dimensional Measurement-space (LoDiM)State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.6.1 Principle of Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

ix

3.6.2 Measurement Selection Procedure . . . . . . . . . . . . . . . . . . . . 71

3.6.3 LoDiM Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 75


3.7 My Method: Reduced Measurement-space Dynamic State Estimation(ReMeDySE) in Power Systems . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.7.1 The Power System Dynamic State Space . . . . . . . . . . . . . . . . . 90


4 Adaptive and Robust Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98



4.4 Traditional Bad Data Detection and Identification . . . . . . . . . . . . . . . . 101

4.4.1 Chi-squares χ2-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.4.2 Largest Normalized Residual Test . . . . . . . . . . . . . . . . . . . . 103

4.5 My Method: Adaptive Kalman Filter with Inflatable Noise Variances(AKF with InNoVa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.5.1 System modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.5.2 Adaptive Kalman Filter Algorithm . . . . . . . . . . . . . . . . . . . . 105

4.5.3 Algorithm Performance Evaluation . . . . . . . . . . . . . . . . . . . . 110

4.5.4 PMU Measurement Evaluation . . . . . . . . . . . . . . . . . . . . . . 112


4.6 My Method: A Two-stage Kalman Filter Approach for Robustand Real-time Power Systems State Tracking . . . . . . . . . . . . . . . . . . 135

4.6.1 Stage One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

x

4.6.2 Stage Two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136


5 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

xi

LIST OF TABLES

2.1 Observability Checking for the 3-machine 9-bus System . . . . . . . . . . . . . 34

2.2 Observability Checking for the 16-machine 68-bus System . . . . . . . . . . . 36

3.1 Kalman Filter Computational Cost Upper Bound . . . . . . . . . . . . . . . . . 69

3.2 Power Method Computational Cost Upper Bound . . . . . . . . . . . . . . . . 72

xii

LIST OF FIGURES

2.1 The two-port π-model of a transmission line . . . . . . . . . . . . . . . . . . . 19

2.2 The 3-machine 9-bus system . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3 The certainty analysis plot of case 1 from Table 2.1Observability Checking forthe 3-machine 9-bus Systemtable.2.1, in the 3-machine 9-bus system . . . . . . 35



2.6 The 16-machine 68-bus system . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.7 The certainty analysis plot of case 1 from Table 2.2Observability Checking forthe 16-machine 68-bus Systemtable.2.2, in the 16-machine 68-bus system . . . 37



2.10 PMUs installed at buses 1, 6 and 8 . . . . . . . . . . . . . . . . . . . . . . . . 39




2.14 A comparison of steady-state certainties for the four “optimal" solutions, basedupon three criteria: maximum (max), minimum (min) and average (mean). . . . 41

2.15 Steady-state estimation uncertainty versus rotor angles for the ideal case, i.e.when PMUs are installed on all buses. . . . . . . . . . . . . . . . . . . . . . . 48

2.16 Steady-state estimation uncertainty versus rotor angles for PMUs installed atbuses 1, 6 and 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

xiii




2.20 The comparison of the four “optimal" solutions based upon three criteria: max,min and mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1 The operation of a discrete Kalman filter cycle . . . . . . . . . . . . . . . . . . 62

3.2 Complexity-Speed-Change Cycle [1] . . . . . . . . . . . . . . . . . . . . . . . 67

3.3 The LoDiM algorithm flow chart. . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.4 Bus 2 Voltage magnitude estimation using conventional Kalman filter . . . . . 79

3.5 Bus 2 Voltage magnitude estimation using randomly chosen measurements . . . 80

3.6 Bus 2 Voltage magnitude estimation using LoDiM . . . . . . . . . . . . . . . . 81

3.7 The performance comparison of state estimation with full measurement-space(in red), with randomly selected measurements (in green) and with LoDiM (inblue), from t = 4 to t = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.8 Bus 60 Voltage magnitude estimation using conventional Kalman filter . . . . . 83

3.9 Bus 60 Voltage magnitude estimation using randomly chosen measurements . . 84

3.10 Bus 60 Voltage magnitude estimation using our LoDiM method for measure-ment selection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.11 The performance comparison of different methods at generator bus 60 in the16-generator 68-bus system, with full measurement-space (in red), with ran-domly selected measurements (in green) and with LoDiM (in blue), from t = 4to t = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.12 The performance comparison of different methods at generator bus 26 in the16-generator 68-bus system, with full measurement-space (in red), with ran-domly selected measurements (in green) and with LoDiM (in blue), from t = 4to t = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

xiv

3.13 Performance comparison of LoDiM with different mσ values, at bus 2 in the3-generator 9-bus system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.14 Performance comparison of LoDiM with different mσ values, at bus 26 in the16-generator 68-bus system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.15 LoDiM performance versus different amount of selected measurements in the3-generator 9-bus system, with 10 megaFLOPS computing power available . . 89

3.16 LoDiM performance versus different amount of selected measurements in the16-generator 68-bus system. with 1 gigaFLOPS computing power available . . 89

3.17 Estimation of rotor speed at generator 2 using ReMeDySE and regular EKF,with a 3-phase fault at bus 29 . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.18 Performance comparison of ReMeDySE and regular EKF estimating rotor speedat generator 2, from t = 2 to t = 5 . . . . . . . . . . . . . . . . . . . . . . . . 95

3.19 Performance comparison of ReMeDySE and regular EKF estimating rotor an-gle at generator 2, from t = 2 to t = 5 . . . . . . . . . . . . . . . . . . . . . . 95

3.20 Performance comparison of ReMeDySE and regular EKF estimating rotor speedat generator 10, from t = 2 to t = 5 . . . . . . . . . . . . . . . . . . . . . . . 95

3.21 Performance comparison of ReMeDySE and regular EKF estimating rotor an-gle at generator 10, from t = 2 to t = 5 . . . . . . . . . . . . . . . . . . . . . . 95

4.1 High-level diagram of Kalman filter, courtesy of Greg F. Welch. System modelerror, and/or measurement error, can all lead to significant state estimation errorwhen passing through a Kalman filter . . . . . . . . . . . . . . . . . . . . . . . 107

4.2 The circular boundary of a 1% TVE criterion . . . . . . . . . . . . . . . . . . . 113

4.3 Voltage estimation results of three filters in case 1 . . . . . . . . . . . . . . . . 116

4.4 Voltage estimation accuracy of three filters in case 1 . . . . . . . . . . . . . . . 117





xv





4.13 Bus 1 voltage estimation results of three filters in case 6 . . . . . . . . . . . . . 132

4.14 Bus 4 voltage estimation results of three filters in case 6 . . . . . . . . . . . . . 133


4.16 Voltage estimation results of three approaches in case 1 . . . . . . . . . . . . . 144

4.17 Voltage estimation accuracy of three approaches in case 1 . . . . . . . . . . . . 145

4.18 Rotor speed and angle estimation results of three approaches in case 1 . . . . . 146

4.19 Rotor speed estimation accuracy of three approaches in case 1 . . . . . . . . . 147

4.20 Rotor angle estimation results of three approaches in case 1 . . . . . . . . . . . 148












xvi










xvii

CHAPTER 1

INTRODUCTION

1.1 Motivations and Topics

From electrical power grids to economics, from microcontroller chips to spacecrafts, sys-

tems are everywhere and a part of everything. Either complex or simple, a system at a given

time point can be determined by a set of system variables. The smallest possible such set is

known as the internal state, or simply the state. Because the set of state variables can fully

specify the system at any give time, it is an essential element in system analysis and control.

While the true values of state variables may never be known exactly, they can be estimated

using the available measurements and established state models. From an engineering per-

spective, numerous professionals have devoted their efforts to developing advanced measuring

technologies and data processing technologies, so that measuring and computing devices are

constantly developing towards higher accuracy and greater efficiency.

Other than designing better measuring instruments, processing data on larger and faster

computational facilities, what else can we do to improve the estimation qualities? Here we

propose three topics from a computational and system-wide perspective:

• Grid Sensor Placement. When the measurement sensors are expensive/limited, it is im-

portant to install them at the most informative locations throughout the system network,

so that they can provide measurement data with optimal observability and reliability.

• Filter Computation Adaptation. When the massive measurement data processing and

1

limited computational resources become obstacles in real-time state estimation, we hope

to reduce the computational complexity of the estimator. Strategic reduction of the num-

ber of measurements used per estimate allows us to do so without sacrificing the estima-

tion quality.

• Adaptive and Robust Estimation. In a hostile environment where the hypothetical models

do not match the actual systems, and/or the measurements contain significant errors, the

estimated states could deviate from the true states rapidly. The influenced estimation has

to be corrected as much as possible.

Our research on these topics, with applications in power systems, can serve as a bridge between

Computer Science and Power Engineering. Furthermore, they can be extended to other large-

scale complex systems as well. The background, previous work and our proposed ideas related

to these three topics will be explained in the following subsections.

1.1.1 Grid Sensor Placement

In our particular project we are dealing with advanced measurement sensors across the

power grid. Traditionally, power grid measurements are provided by remote terminal units

(RTUs) at the substations. RTU measurements include real/reactive power flows, power injec-

tions, and magnitudes of bus voltages and branch currents.

A novel phasor measurement technology was developed in the 1980s [2], [3]. The crucial

devices, called phasor measurement units(PMUs), are able to measure both magnitude and

phase angle of the electrical waves in a power grid. With the global positioning system (GPS),

PMU provide high-speed and high-accuracy sensor data with precise time synchronization.

These synchronized phasor measurements are commonly referred to as synchrophasors. As a

comparison, RTU measurements are much slower and are not well synchronized.

2

While PMU measurements currently cover fewer than 1% of the nodes in the U.S. power

grid, the power industry actively invests in advancing the technology and installing more units.

However at the present stage, the installation must be selective because of resource constraints.

Previous PMU placement research has focused primarily on network topology, with the

goal of finding configurations that achieve full network observability with a minimum number

of PMUs [4–7]. However we have noticed two issues about this problem: 1) The observability

of the entire system is not an arall or nothingas problem. PMUs are added to the grid incre-

mentally, and it will be useful to have some guidance on where to install them. 2) The power

system networks usually have complex topologies, so that when we obtain more than one “op-

timal placement" solution with the same number of PMUs, one will have to make a selection

among these solutions.

To address these issues, recently we introduced an evaluation method that utilizes stochastic

models of the signals and measurements to characterize the observability and corresponding

uncertainty of power system states, for any given configuration of PMUs [8]. This evaluation

method includes a novel observability checking algorithm and estimation uncertainty analysis.

The observability checking classifies all the observable buses into one or more levels, according

to the ardirectnessas of their observations: Level 1 contains the most directly observable buses

(PMU buses), while the buses with more “indirect" (Section 2.8.1) observations are labeled

with higher levels. Then based on the previous work [9, 10], we compute a stochastic estimate

of the asymptotic or steady-state error covariance, P∞. P∞ is used as a quantitative metric

for the performance evaluation. We generalize our approach for any PMU placement layout

without requirements for achieving full observability.

Because PMUs can provide real-time synchrophasor data to the Supervisory Control And

Data Acquisition (SCADA) system to capture the dynamic characteristics of the power system,

they have been found useful in estimating more transient states [11]. Thus we have extended

our evaluation method to both static (bus voltage magnitudes and phase angles) and dynamic

3

(generator rotor speed and rotor angles) state estimation applications. In [12], a new approach

is presented to design an optimal PMU placement according to estimation uncertainties of the

power system dynamic states. We restate the optimal PMU placement problem so that the so-

lution is not only a minimum set of PMUs that can cover the entire power system, but also

the one helps dynamic state estimation the most. So if an integer programming solver returns

multiple placement solutions, which all achieve full observability with the same minimum

number of PMUs, we apply steady-state uncertainty analysis to evaluate the solutions. The op-

timal PMU placement is hence selected according to the evaluation results. This approach can

provide planning engineers with a new tool to help in the selection between PMU placement

alternatives.

1.1.2 Filter Computation Adaptation

Dynamic state estimation (DSE) techniques, such as the Kalman filters [13], have been the

subject of extensive research and applications. Especially in modern power systems, DSE is a

fast developing research area. Compared to traditional steady state estimators, DSE techniques

are able to track dynamic state variables both efficiently and accurately.

However, in large-scale and wide-area inter-connected power systems, the combination of

high computational complexity and slow processing speed presents a significant challenge. To

help address this challenge, it is a good idea to employ faster response filters with reduced

measurement-space.

In our recent work we have set our sight on measurement-space reduction: we attempt to

select and process the measurement subsets which can improve state calculations the most. As

inspired by the previous single-constraint-at-a-time (SCAAT) method [14], an approach we call

Lower Dimensional Measurement-space (LoDiM) state estimation was developed in [15, 16].

4

The LoDiM approach employs an estimator such as the Kalman filter, but with a twist:

while the filter is running in the foreground, it also performs a measurement selection procedure

in the background to optimally reduce the measurement dimension. The measurement selection

procedure utilizes the principal component analysis (PCA) of the error covariance, to determine

the subset of the state space to update most urgently (e.g. with larger estimation uncertainties

than others). We prefer to target at the principal uncertainties first, thus the measurements

are ranked according to their abilities to calibrate these uncertainties. Only a smaller subset

that can reduce the principal uncertainties most efficiently is then processed by the foreground

filter during each cycle. As a result, during each cycle it yields much smaller computational

load and hence lower latency. Furthermore, we have also extended our approach to Reduced

Measurement-space Dynamic State Estimation (ReMeDySE), for estimating the dynamic state

of power systems [17].

Although we present LoDiM in the context of the Kalman filter, the associated measure-

ment selection procedure is not filter-specific, i.e. it can be used with other state estimation

methods, as long as the state uncertainties are dynamically estimated along with the states.

Our approach is not application-specific either: it can be applied to other large-scale, real-time

on-line and computationally intensive state tracking systems with modern parallel computation

techniques.

1.1.3 Adaptive and Robust Estimation

Kalman filters achieve optimal performance only when the system noise characteristics

have known statistics that obey certain properties (zero-mean, Gaussian, and spectrally white).

However in practice, it is difficult to obtain the process and measurement noise models. When

the theoretical models do not match the actual models, and/or the measurements contain sig-

nificant errors, the estimated state can deviate from the true state rapidly. When people notice

5

estimated state deviate from true state, it is usually impossible to determine whether the devia-

tion is caused by a process model mismatch, a measurement model mismatch, or both.

To address this problem, we propose an efficient adaptive approach called the Adaptive

Kalman Filter (AKF) with Inflatable Noise Variances (InNoVa). Primary simulations have

demonstrated the robustness of this approach facing sudden changes of system dynamics and

erroneous measurements, with the ability to adjust noise modeling parameters on-the-fly. It is

capable of doing these because besides the regular normalized innovation test, we also employ

a normalized residual test to help separate the process and measurement factors.

Furthermore, we have considered incorporating AKF with InNoVa into the dynamic state

estimation algorithm from previous research [11], to also help with the estimation of power

system dynamic states. Thus we developed a two-stage Kalman filter approach to estimate the

static states of voltage magnitudes and phase angles, as well as the dynamic states of generator

rotor angles and generator speeds. In the first stage, we estimate the static states from raw

PMU measurements, using AKF with InNoVa, which is able to identify and reduce the impact

of incorrect system modeling and/or bad PMU measurements. Then in the next stage, the

estimated bus voltages are fed into a second extended Kalman filter to obtain the dynamic state

estimation. Case studies show the inherent advantages of our two-stage Kalman filter technique

over the extended Kalman filter alone.

1.2 Thesis Statement

Combinations and comparisons of process and measurement uncertainties can

be used to assess/optimize measurement unit placement, improve computational

efficiency and flexibility, and increase system robustness. Specifically,

1. steady-state uncertainty analysis can be combined with topological analysis

6

to evaluate or optimize measurement unit placement in a network;

2. sensitivity-to-uncertainty ratio ranking can be used to select measurement

subsets to optimally adapt to the available computational resources; and

3. innovation-residual tests can be used to identify apparent outlier measure-

ments, and adapt the process/measurement models.

1.3 Contributions

• Our grid sensor placement approach is able to evaluate and compare any candidate sensor

placement configuration in complex networks, regardless of achieving full observability.

While off-line steady-state uncertainty analysis has aided in optimal sensor placement

for continuous-space systems such as the 3D tracking system, the combination of this

with a network topology analysis enables sensor placement evaluation and comparison

for discrete-space systems such as the power system.

• Our Lower Dimensional Measurement-space (LoDiM) state estimation algorithm, adapt-

ing to the computation budget available, can help many large-scale, real-time on-line and

computation-intensive state tracking systems because

1) it features a dynamic measurement selection procedure: a small measurement

subset is dynamically and strategically chosen for each cycle. The smaller measurement-

space incorporate less information per cycle, but has higher reporting rates;

2) the measurement selection procedure is not filter-specific, i.e. it can be used with

other state estimation methods such as particle and unscented filters.

• Our Adaptive Kalman Filter with Inflatable Noise Variances (AKF with InNoVa) enables

robust state estimation. Its ability of adjusting noise modeling parameters on-the-fly can

efficiently

7

1) identify and adapt to incorrect system modeling;

2) identify and reduce the impact of erroneous measurements.

1.4 Thesis Organization

There are three main topics in this thesis. They are individually studied, yet closely related.

Among which, I investigate uncertainty-aware grid sensor placement stratagies in Chapter 2.

Next, Chapter 3 discusses how to use uncertainty-guided measurement selection approach to

reduce computational burden. Then I present our lately developed adaptive and robust state

estimation algorithms, based on uncertainty analyses, in Chapter 4. Finally, Chapter 5 summa-

rizes our work, and proposes possible plans for the future research.

For each thesis topic, I will state the structure of its own chapter more specifically in the

“Chapter Organization" section.

8

CHAPTER 2

GRID SENSOR PLACEMENT

The synchronized phasor measurement unit (PMU), developed in the 1980s [2, 3], is con-

sidered one of the most important devices in the future of power systems. While PMU measure-

ments currently cover fewer than 1% of the nodes in the U.S. power grid, the power industry

has gained the momentum to advance the technology and install more units. However, with

limited resources, the installation must be selective.

Previous PMU placement research has focused primarily on the network topology, with the

goal of finding configurations that achieve full network observability with a minimum number

of PMUs. Here we introduce a new approach that also includes stochastic models for the

signals and measurements, to characterize the observability and corresponding uncertainty of

any given configuration of PMUs, whether that configuration achieves full observability or not.

Utilization of this approach to designing optimal PMU placements, according to estimation

uncertainties of the power system static states/dynamic states, are also discussed in details.

This approach can provide planning engineers with a new tool to help choose between PMU

placement alternatives.

2.1 Introduction

State estimation plays a crucial role in determining the health of a power system. State

can be estimated using the available measurements and system model. Traditionally, remote

terminal units (RTU) at the substations provide power grid measurements. The most commonly

used RTU measurements in state estimation are:

9

• Line power flow measurements: the real and reactive power flow along the transmission

lines or transformers.

• Bus power injection measurements: the real and reactive power injected at the buses.

• Voltage magnitude measurements: the voltage magnitudes of the buses.

Under certain circumstances such as state estimation of distribution systems, the line current

magnitude measurements may be considered, which provide the current flow magnitudes along

the transmission lines or transformers.

With the ability to impact future power system monitoring and operation methods, phasor

measurement technology is one of the key enabling technologies for the future “smart grid".

Phasor measurement units (PMUs) make two additional types of measurements available:

• Voltage phasor measurements: the phase angles and magnitudes of voltage phasors at

system buses.

• Current phasor measurements: the phase angles and magnitudes of current phasors along

transmission lines or transformers.

Recent development of phasor measurement technologies offers high-speed sensor data (typi-

cally 30 or 60 samples/second) with precise time synchronization. This is in comparison with

traditional RTU measurements in SCADA system, which have cycle times of 2 to 4 seconds

and are not well synchronized.

PMUs are becoming increasingly attractive in various power system applications such as

system monitoring, protection, control, and stability assessment. Widely scattered in the power

system network and synchronized from the common global positioning system (GPS) radio

clock, PMUs can provide real-time synchrophasor data to the SCADA system to capture the

dynamic characteristics of the power system, and hence facilitate time-critical applications

10

such as dynamic state estimation and dynamic stability analysis [11]. Compared to estimating

relatively stationary state elements such as bus voltage magnitudes and phase angles, dynamic

state estimation seeks to estimate more transient states of a power system, i.e. generator rotor

angles and speeds. With properly placed and sufficient numbers of PMUs, we can estimate the

system status in real-time using time stamped synchrophasors.

There has been significant work in selecting the best locations to install PMUs [4–7]. Many

algorithms have been developed, primarily with the aim of utilizing a minimum number of

PMUs to ensure full network observability. However, two issues caught our attention:

• First, the observability of the entire system is not an “all or nothing” problem. Due to

various resource limitations, we do not have the luxury of installing a complete set of

PMUs throughout the power grid. Yet we are not starting from scratch either. In practice

people are installing PMUs to the grid incrementally, thus it would be useful to have

some guidance on where to install them.

• Second, in the previous research, the network topology was the primary focus. In practice

the networks usually have complex topologies where more than one solution for the same

minimum number of PMUs will appear. In such cases, the planning engineers need to

make a choice from these solutions.

We focus on bridging these gaps. Specifically our contributions include:

• We develop a stochastic model that captures state estimation uncertainties, to facilitate

PMUs installation assessment.

• We design an optimal PMU placement evaluation algorithm by incorporating uncertainty

estimates into topological considerations for the specific network.

• We present an approach to the comparison among alternative configurations via quanti-

tative measure of expected uncertainties.

11

2.2 Chapter Organization

The rest of this chapter is organized as follows. Section 2.3 introduces the background. In

Section 2.4 and Section 2.5, I review the basic concepts of state space models and steady state

estimation briefly. Section 2.6 discusses traditional power system state estimation method.

Section 2.7 describes the PMU devices and their impact on power system state estimation.

Then for static (Section 2.8) and dynamic (Section 2.9) state estimation respectively, we present

our approach to determine the effects of any given PMU configuration, and restate the optimal

PMU placement problem such that the solution is not only a minimum set of PMUs that can

cover the entire power system, but also the set that improves state estimation the most. Our

methods are simulated on multi-machine system models to demonstrate how it might help

engineers make decisions on different candidate plans.

2.3 Background and Related Work

The phasor measurement unit (PMU) was first developed and utilized in [2, 3]. Consid-

ering partially observable systems (with an inadequate number of PMUs) the authors in [18]

presented an estimation algorithm based on singular value decomposition (SVD), which did

not require fully observable system prior to estimation.

Optimal PMU placement for full observability was studied in [4]. They developed an algo-

rithm for finding the minimum number of PMUs required for power system state estimation,

using simulated annealing optimization and graph theory in formulating and solving the prob-

lem.

In [7] the authors focused on the analysis of network observability and PMU placement

when using a mixed measurement set. They developed an optimal placement algorithm for

12

PMUs using integer programming. In [5], a strategic PMU placement algorithm was developed

to improve the bad data processing capability of state estimation by taking advantage of the

PMU technology. Furthermore, [6] re-studied the PMU placement problem and presented a

generalized integer linear programming formulation.

Our work, however, is taking a different approach: we first characterize the observability

and corresponding uncertainty of the power system buses for any given configuration of PMUs,

whether that configuration achieves full observability or not. Then we also presented our work

on connecting the optimal PMU placement with power system static as well as dynamic state

estimation: we provide a method for evaluating any candidate PMU placement design, which

is especially useful when there are multiple placement candidates [8], [12].

2.4 State Space Models

The state space models are the most basic yet extensively used mathematical models in

all kinds of state estimation problems. Here I present a brief review to introduce the relevant

mathematical notations used throughout this dissertation. For more detailed description, the

readers can refer to [19].

An assumed linear system can be modeled as a set of linear stochastic process equation

and measurement equation:

xk = Axk−1 +Buk−1 + wk−1 (2.1)

zk = Hxk + vk (2.2)

where x ∈ Rn is the state vector, A is a n× n matrix that relates the state at the previous time

step k − 1 to the state at the current step k, without either a driving function or process noise1,

u ∈ Rl is the optional control input vector. B is a n × l matrix that relates the control input1In practice, the matrix A may change with each time step, but it is assumed to be constant here.

13

vector to the state xk, z ∈ Rm is the measurement vector, H is a m × n matrix that relates

the state xk to the measurement zk. w ∈ Rn and v ∈ Rm are process and measurement noise

vectors respectively.

The process noise w and measurement noise v are assumed to be mutually independent

random variables, white, and with normal probability distributions:

p(w) ∼ N(0, Q) (2.3)

p(v) ∼ N(0, R) (2.4)

where the process noise covariance Q and measurement noise covariance R matrices are usu-

ally assumed to be constant.

In reality, the process to be estimated and (or) the measurement relationship to the process

is usually non-linear. Especially when our objective is to estimate the dynamic states of the

power system. A non-linear system can be modeled using non-linear stochastic process and

measurement equations

xk = a(xk−1, uk−1) + wk−1 (2.5)

zk = h(xk) + vk. (2.6)

where x ∈ Rn is the state vector, a is a non-linear function that relates the state at the previous

time step k−1 to the state at the current step k. u is the (optional) driving function, considered

as parameters in function a. z ∈ Rm is the measurement vector. h is another non-linear

function that relates the state xk to the measurement zk. w and v again represent the zero-mean

process and measurement noise as defined before.

These non-linear functions can be linearized about the point of interest x in the state space.

To do so one need to compute the state Jacobian and measurement Jacobian matrices

A =∂a(x)

∂x|x (2.7)

H =∂h(x)

∂x|x (2.8)

14

where A and H are the partial derivatives of a and h, with respect to x.

For any state estimate xk, we define the estimate error as ek ≡ xk − xk and estimate error

covariance as Pk ≡ E[ekeTk ], where E denotes statistical expectation. For the state estimate xk

at the time step k, the estimate error covariance Pk contains important information, reflecting

the uncertainty of the estimation.

2.5 Steady State Estimation

On-line methods such as the Kalman filter [13, 19] can be used to estimate time varying

state and error covariance, by minimizing the a posteriori estimate error covariance Pk in a

recursive predictor-corrector fashion. More details can be found in Section 3.4. In these on-

line scenarios the estimate error covariance Pk changes over time. However the steady-state

error covariance

P∞ = limk→∞

E[(xk − xk)(xk − xk)T ]

= limk→∞

E[ekeTk ] (2.9)

can be computed off-line in closed form. Here x and x represent the true and estimated states

respectively, and E denotes statistical expectation. In fact to compute the steady-state uncer-

tainty one does not actually need to estimate xk and xk. Instead one can estimate P∞ directly

from state-space models of the system using stochastic models for the various noise sources.

The rest of this section reviews the method for estimating P∞ based on previous work

[9, 10], which employs the Discrete Algebraic Riccati Equation (DARE). The DARE represents

a closed-form solution to the steady-state covariance P∞ [20]. Assuming the process and

measurement noise elements are uncorrelated, it can be written in this form:

P∞ = AP∞AT +Q− AP∞HT (R +HP∞HT )−1HP∞AT (2.10)

15

The derivation of the DARE will be explained in Section 3.4.1.

The DARE solution P∞ can be calculated using the MacFarlane-Potter-Fath “Eigenstruc-

ture Method” [20]. Given the model parameters A, Q, H , and R from the last subsection, the

2n× 2n discrete-time Hamiltonian matrix is written as

Ψ =

A+QA−THTR−1H QA−T

A−THTR−1H A−T

. (2.11)

Then after calculating n characteristic eigenvectors [e1, e2, ..., en] of Ψ, we form a matrix B

C

= [e1, e2, ..., en] (2.12)

Finally, we use B and C to compute the steady-state error covariance

P∞ = BC−1. (2.13)

Note that if the Jacobians A and H are functions of the state, we will generally have to compute

them at each point of interest in the state space. The noise covariance matrices R and Q might

be constant, or might also vary as a function of the state.

For each point of interest in the state space, P∞ indicates the expected asymptotic state

estimation uncertainty corresponding to the candidate design modeled by the specific A, Q, H ,

and R. Intuitively, given PMU placements leading to the same level of observability, the lower

the overall uncertainty is, the more we prefer this design.

2.6 Traditional Power System State Estimation

Research in power system state estimation can be traced back to the late 1960ars. The

conventional state estimator provides the estimates of the power system static states, i.e. bus

voltage phasors (magnitudes and phase angles), based on measurements obtained from the

16

SCADA system. Usually, people use weighted least squares (WLS) estimators to find the best

estimates of the states. In this subsection, we give a brief review of traditional power system

state estimation method and model, based on the classic textbook [21]. We refer this procedure

as a static state estimation because it obtains the voltage phasors at all buses at a single point

in time. In other words, WLS estimator only focuses on the measurement equations (2.2, 2.6)

in the state space model, without considering the process equations (2.1, 2.5).

2.6.1 Weighted Least Squares Estimator

The measurements used by the estimator are the real and reactive power flows and bus

injections, and the voltage magnitudes |V |, as provided by SCADA data. Traditionally the

phase angles, θ, are not known. It is assumed that bad data have been eliminated from this

measurement set by the usual methods. The measurements are usually formulated in vector

notation as

z = h(x) + v (2.14)

where z is the measurement vector, x is the state vector, v is the vector of the measurement er-

rors from noise, and the vector function h is the nonlinearity between the power measurements

and the state.

The WLS estimator aims at minimizing the objective function J , which is formulated as

the sum of the squares of the weighted residuals:

J(x) = [z − h(x)]TR−1[z − h(x)] (2.15)

where R is the measurement error covariance matrix. At the kth iteration of the WLS solution,

the residual vector r is defined as

rk = z − h(xk). (2.16)

17

To minimize the objective function J , the first-order optimality conditions must be satisfied,

that is

g(x) =∂J(x)

∂x= −HT (x)R−1[z − h(x)] = 0, (2.17)

where

H(x) =∂h(x)

∂x(2.18)

is the Jacobian matrix obtained by taking partial derivatives of h with respect to x.

A non-linear optimization algorithm, such as the Newton-Raphson technique, is used to

iteratively converge to a solution. Assuming the Newton-Raphson method and the first-order

Taylor expansion are applied, along with an initial value of x, the step at each iteration k

becomes:

xk+1 = xk + [G(xk)]−1HT (xk)R−1[z − h(xk)], (2.19)

where G(xk) is the gain matrix given by

G(xk) =∂g(xk)

∂x= HT (xk)R−1H(xk). (2.20)

The gain matrix is positive definite and symmetric provided that the system is fully observable,

and it is normally sparse.

The estimator is considered to have converged when the reduced-order measurement mis-

match vector HT (xk)R−1[z − h(xk)] has reached some prescribed threshold.

2.6.2 The Measurement Function and Measurement Jacobian

As mentioned before, the most commonly used measurements provided by traditional

SCADA system are the line power flows, bus power injections, bus voltage magnitudes, and

18

sometimes line current magnitudes. These measurements can be expressed in terms of the state

variables either using the rectangular or the polar coordinates—the conventional choice is polar

coordinates. Here we briefly describe the measurement equations provided by [21].

If a system contains N buses, the state vector will have (2N − 1) elements: N bus voltage

magnitudes and (N − 1) bus voltage phase angles, as one bus is chosen as the reference bus

with its phase angle set equal to an arbitrary value such as 0. This bus is also known as the

slack bus or swing bus.

Typically we choose bus 1 as the reference bus. The state vector will have the following

form:

x = [θ2θ3...θNV1V2...VN ]T . (2.21)

Assuming the general two-port π-model for the power system transmission lines (as shown

in Fig. 2.1),

Figure 2.1: The two-port π-model of a transmission line

the above mentioned measurements can be expressed in the following equations:

19

• Real power injection at bus i:

Pi = Vi

N∑j=1

Vj(Gij cos θij +Bij sin θij) (2.22)

• Reactive power injection at bus i:

Qi = Vi

N∑j=1

Vj(Gij sin θij −Bij cos θij) (2.23)

• Real power flow from bus i to bus j:

Pij = V 2i (gsi + gij)− ViVj(gij cos θij + bij sin θij) (2.24)

• Reactive power flow from bus i to bus j:

Qij = −V 2i (bsi + bij)− ViVj(gij sin θij − bij cos θij) (2.25)

• Line current flow magnitude from bus i to bus j:

Iij =

√P 2ij +Q2

ij

Vi

(2.26)

Because in practice the shunt admittance gsi + jbsi is much smaller comparing to the

series admittance gij + jbij , the shunt admittance is usually ignored:

Iij =√(g2ij + b2ij)(V

2i + V 2

j − 2ViVj cos θij) (2.27)

• Voltage magnitude at bus i:

Vi = Vi (2.28)

where

Vi is the voltage magnitude at bus i,

θi is the phase angle at bus i,

20

θij = θi − θj is the phase angle difference between buses i and j,

gsi + jbsi is the shunt admittance connected at bus i,

gij + jbij is the series admittance of the transmission line connecting buses i and j,

Gij + jBij is the ijth element of the complex bus admittance matrix.

The measurement Jacobian matrix H then has the following form:

H =

∂Pinj

∂θ

∂Pinj

∂V

∂Pflow

∂θ

∂Pflow

∂V

∂Qinj

∂θ

∂Qinj

∂V

∂Qflow

∂θ

∂Qflow

∂V

∂Imag

∂θ

∂Imag

∂V

0 ∂Vmag

∂V

. (2.29)

where

Pinj and Qinj are the real and reactive power injection measurements,

Pflow and Qflow are the real and reactive power flow measurements,

Imag and Vmag are the line current flow magnitude and voltage magnitude measurements.

The elements of this measurement Jacobian matrix H can be derived from the measurement

equations conveniently:

21

• Partial derivatives of real power injection at bus i:

∂Pi

∂θi=

N∑j=1

ViVj(−Gij sin θij +Bij cos θij)− V 2i Bii (2.30)

∂Pi

∂θj= ViVj(Gij sin θij −Bij cos θij) (2.31)

∂Pi

∂Vi

=N∑j=1

Vj(Gij cos θij +Bij sin θij) + ViGii (2.32)

∂Pi

∂Vj

= Vi(Gij cos θij +Bij sin θij) (2.33)

• Partial derivatives of reactive power injection at bus i:

∂Qi

∂θi=

N∑j=1

ViVj(Gij cos θij +Bij sin θij)− V 2i Gii (2.34)

∂Qi

∂θj= ViVj(−Gij cos θij −Bij sin θij) (2.35)

∂Qi

∂Vi

=N∑j=1

Vj(Gij sin θij −Bij cos θij)− ViBii (2.36)

∂Qi

∂Vj

= Vi(Gij sin θij −Bij cos θij) (2.37)

• Partial derivatives of real power flow from bus i to bus j:

∂Pij

∂θi= ViVj(gij sin θij − bij cos θij) (2.38)

∂Pij

∂θj= −ViVj(gij sin θij − bij cos θij) (2.39)

∂Pij

∂Vi

= −Vj(gij cos θij + bij sin θij) + 2(gij + gsi)Vi (2.40)

∂Pij

∂Vj

= −Vi(gij cos θij + bij sin θij) (2.41)

22

• Partial derivatives of reactive power flow from bus i to bus j:

∂Qij

∂θi= −ViVj(gij cos θij + bij sin θij) (2.42)

∂Qij

∂θj= ViVj(gij cos θij + bij sin θij) (2.43)

∂Qij

∂Vi

= −Vj(gij sin θij − bij cos θij)− 2(bij + bsi)Vi (2.44)

∂Qij

∂Vj

= −Vi(gij sin θij − bij cos θij) (2.45)

• Partial derivatives of line current flow magnitude from bus i to bus j, shunt admittance

ignored:

∂Iij∂θi

=g2ij + b2ij

IijViVj sin θij (2.46)

∂Iij∂θj

= −g2ij + b2ij

IijViVj sin θij (2.47)

∂Iij∂Vi

=g2ij + b2ij

Iij(Vi − Vj cos θij) (2.48)

∂Iij∂Vj

=g2ij + b2ij

Iij(Vj − Vi cos θij) (2.49)

• Partial derivatives of voltage magnitude at bus i:

∂Vi

∂Vi

= 1 (2.50)

∂Vi

∂Vj

= 0 (2.51)

∂Vi

∂θi= 0 (2.52)

∂Vi

∂θj= 0 (2.53)

2.7 The Phasor Measurement Units (PMUs)

Synchronized by GPS satellite clock, PMUs have achieved a level of precision that typically

exceeds the conventional measurements, which makes phasor telemetry a valuable source of

23

measurements. Nowadays, PMUs are becoming more and more attractive in various power

system applications such as system monitoring, protection, control, and stability assessment.

The benefit of PMUs is especially evident when it comes to power system state estimation.

Conventional power system state estimation uses real and reactive power as measurements

to estimate the bus phasor voltages. As a result, the relationship between measurements and

states is non-linear. The solution is always obtained by linearizing the model and solving it in

an iterative fashion.

PMUs alleviate this problem by providing the phasors of voltages and currents measured

at a given substation. Hence, by measuring the bus phasor voltages and line phasor currents,

the relationship between measurements and states becomes linear. When the measurement

equation is linear, the estimation algorithm is direct and significantly faster than the non-linear

ones.

2.7.1 Observability Checking

For modeling power systems, we consider two types of measurements: PMU measurements

and zero injection measurements.

A PMU installed at a specific bus is capable of measuring not only the bus voltage phasor,

but also the current phasors along all the lines incident to the bus. Hence in addition to the

phasor voltage at this bus, we are able to compute the phasor voltages of all of its neighboring

buses.

If a bus has neither generators nor loads, it is a zero injection bus [21]. The inclusion of

zero injection buses has been proven effective in reducing the number of PMUs needed for an

optimal placement plan [7]. Because the sum of flows on all the associated branches to the bus

is zero, this equality normally corresponds to a pseudo measurement called the zero injection

24

measurement [7], which is useful in system state estimation. According to Kirchhoff’s current

law, we know that at any bus in the power grid, the sum of currents flowing into that bus is

equal to the sum of currents flowing out of that bus. Thus, for an injection-measured bus and

its n neighbors ((n+1) buses total), if the phasor voltages at any n out of the (n+1) buses are

known (observable), then the remainder can also be computed and hence become observable.

Consistent with previous studies, we only consider isolated zero injection buses in our test

systems. Meaning that, once a zero injection bus location is used in our algorithm, other zero

injection buses adjacent to it (if there is any) will not be taken into account. This is because

although zero injection buses can help reduce the PMUs needed for a system, the reduction of

PMUs will compromise the system reliability. More detailed explanation can be found in [22].

Based on these assumptions, we have developed the observability checking algorithm shown

in Algorithm 1.

2.8 My Method: PMU Placement for Power System Static State Estima-

tion

2.8.1 PMU Placement Evaluation

Previous studies consider a PMU placement optimal in the sense that it maximizes observ-

ability while minimizing cost. The authors in [6, 7] presented a numerical formulation of this

problem, and developed an optimal placement algorithm for PMUs using integer linear pro-

gramming. This formulation allows easy analysis of the network observability and concerns

about the full coverage of the system. However, the integer linear programming approach may

be insufficient for determining the optimal locations of PMUs. The reason is that very often

there will be multiple optimal solutions with the same minimum number of PMUs. In such

25

Algorithm 1: Observability checking for a given set of PMU locations and zero injectionbuses.

Input: PMU buses location P_bus and zero injection buses location Z_busOutput: Observable buses location O_busO_bus = ∅;O_bus = P_bus (the buses that can be measured directly by PMUs);for i← 1 to |P_bus| do

O_bus = O_bus ∪ neighbors(P_busi)(the | · | operator denotes the cardinality of the set);(adding the buses that can be estimated through at least one adjacent PMU bus);flag = 1;while flag do

flag = 0;for j ← 1 to |Z_bus| do

if(|Z_busj ∪neighbors(Z_busj))∩O_bus| = |Z_busj ∪neighbors(Z_busj)|−1then

O_bus = O_bus ∪ Z_busj ∪ neighbors(Z_busj);flag = 1;(adding the buses that can be estimated using the zero injection information.According to the Kirchhoff’s law: for a zero injection bus and itsneighborhood, if all but one is known, then the unknown one can beestimated );

cases, one would need a means for comparing the different solutions, as some will actually

offer lower estimate uncertainty and latency.

Built on the previous work by Allen et al. [9, 10], our approach is to use a stochastic

estimate of the asymptotic or steady-state error covariance, as a quantitative metric for the

performance evaluation. Our approach is generalized for any PMU placement layout, without

requirements for achieving full observability.

To estimate the steady-state error covariance, we first consider the relevant state space

models.

26

The Process Model

For modeling the state and covariance changes over time, there are several ways to iden-

tify the parameters A and Q, both a priori and on-line [10, 20]. Following the methodology

described in [23], a quasi-static model of an assumingly stable power system is used in our ap-

proach. The oscillations in the state variables of this model are assumed small, thus the states

of the power system at time step (k + 1) are the same as those at time step k, except for some

zero mean, white Gaussian noise. Hence our process model reduces to

xk = xk−1 + wk−1. (2.54)

The Measurement Model

Traditionally, real and reactive power measurements are used in the power system state

estimation, resulting in non-linear measurement models and iterative algorithm to solve for the

state of the system. Now with the help of PMUs, the measurement model becomes linear: we

simply let the phasor bus voltages be the state, while phasor bus voltages and phasor currents

be the measurements. In this way we are able to use non-iterative estimation algorithms, which

speeds up the computation signficantly.

The measurement model has the general form

z = Hx+ v, (2.55)

where x ∈ Rn is the complex state vector, z ∈ Rm is the complex measurement vector, and

H is a m × n complex matrix that relates the state to the measurement zk. The measurement

noise elements in v are assumed to be mutually independent random variables, white, and with

normal probability distributions

p(v) ∼ N(0, R). (2.56)

27

For computational convenience, we convert the complex valued measurement model into a real

valued measurement model as in [18]: Re(z)

Im(z)

=

Re(H) −Im(H)

Im(H) Re(H)

Re(x)

Im(x)

+

Re(v)

Im(v)

(2.57)

where Re(·) and Im(·) stand for the real and imaginary parts of the corresponding vector/matrix

respectively.

After the procedure of observability checking, all the observable buses are divided into one

or more levels, according to the “directness” of their observations: with Level 1 containing

the most directly observable buses, and the buses with more “indirect” observations labeled

with higher levels. Note that if the power grid only uses PMU measurements, the observable

buses belong to either Level 1 or Level 2; however, if both PMU measurements and injection

measurements (which could be zero injection pseudo-measurements) are taken into account,

there may be more than two levels of observable buses.

Level 1 buses

If the bus i is in Level 1 (meaning that there is a PMU placed on this bus), the measurement

equation is

zi = xi + vi, (2.58)

where zi is the measured complex voltage at bus i, xi is the “true” complex voltage at bus i and

vi is the complex measurement noise of this PMU. Thus for all the buses in Level 1, we can

write the measurement equation in the matrix form

zV = I · xL1 + vV , (2.59)

28

where zV is the complex voltage measurement subvector, I is the identity matrix, xL1 is the

complex state subvector (“true” complex voltages at all Level 1 buses) and vV is the voltage

measurement noise subvector.

Level 2 buses

If the bus i is in Level 2 (meaning that there is at least one PMU placed on an adjacent bus),

then for each PMU placed at some adjacent bus j, the measurement equation will be

zji =

(Yji −Yji

) xj

xi

+ vj, (2.60)

where zji is the measured complex current at bus j (towards bus i), Yji is the admittance of

line (j, i), xj and xi are the “true” complex voltages at bus j and i respectively, and vj is the

complex measurement noise of this PMU. So for all the buses in Level 1 and Level 2, we have

the measurement equation

zC =

(YCL1 YCL2

) xL1

xL2

+ vC , (2.61)

where zC is the complex current measurement subvector, YCL1 and YCL2 are the line admit-

tance matrices that relates Level 1 and Level 2 bus voltages to zC respectively, xL1 and xL2

are the complex state subvector (“true” complex voltages at all Level 1 and Level 2 buses),

and vC is the current measurement noise subvector. In this equation, the measurement matrix(YCL1 YCL2

)has each row sum up to zero. If the number of PMUs installed is sufficient,

or the power grid is well-connected, it is quite possible that a Level 2 bus has more than one

PMU-installed neighbor buses. In such cases, the measurement equation has fused the contri-

butions from each adjacent PMU.

29

Level 3 or above buses

If the bus i is in Level 3 or above (meaning that we can compute the bus voltage, but there

is no PMU placed on itself or any adjacent bus), then we must use the information about net

injection current measurements, and Kirchhoff’s Current Law, to calculate the complex bus

voltage. Generally there are two possibilities:

• an injection measurement is taken at bus i; or

• an injection measurement is taken at bus j, which is adjacent to bus i.

Overall, assuming there are totally l levels of observable buses, the measurement equation can

be written as

zI =

(YIL1 YIL2 YIL3 · · · YILl

)

xL1

xL2

...

xLl

+ vI , (2.62)

where zI is the complex injection current measurement subvector (zero pseudo-measurements),

YIL1 through YILlare the node admittance matrices of all observable buses that are related to

zI , xL1 through xLlrepresent the complex state subvector for all observable buses, and vI is

the current measurement noise subvector.

30

Combining the above three cases, the complete measurement model can be expressed as

zV

zC

zI

=

I 0 0 · · · 0

YCL1 YCL2 0 · · · 0

YIL1 YIL2 YIL3 · · · YILl

xL1

xL2

...

xLl

+

vV

vC

vI

. (2.63)

The measurement model can be converted into real valued model, using the technique men-

tioned before.

2.8.2 Optimal PMU Placement

Conventionally, the PMU placement is considered optimal in the sense that it makes all

buses in the system observable with a minimum number of PMUs. This is especially true for

large-scale complex systems. The authors in [6, 7] presented a numerical formulation of this

optimization problem:

min

n(buses)∑i

wi ·Xi (2.64)

s.t. f(X) = C ·X ≥ 1

where n(buses) is the number of buses in the system, wi is the cost of the PMU installed at bus

i, X is a binary decision variable vector with entries Xi defined as

Xi =

1 if a PMU is installed at bus i

0 otherwise, (2.65)

31

1 is a vector whose entries are all ones, f(X) is a vector function and C is the binary connec-

tivity matrix with entries

C[k,m] =

1 if k = m

1 if buses k and m are connected

0 otherwise

. (2.66)

This formulation allows easy analysis of the network observability and concerns about the

full coverage of the system. We first solve this optimization problem using integer linear pro-

gramming techniques. However, the integer linear programming approach is insufficient for

determining the optimal locations of PMUs. The reason is that very often there will be multi-

ple “optimal" solutions with the same minimum number of PMUs. In such cases, one would

need a means for comparing the different solutions, as some will actually offer lower estimate

uncertainty and latency.

Thus the next step is to evaluate the solution(s) using steady-state uncertainty analysis as

described in section 2.5. Naturally, we prefer the solution with lower uncertainties. If there

is no “absolute winner", (i.e. at each bus, the uncertainty value computed for this solution is

lower than any other solution), the user need to define a criterion to judge each solution, and

we will discuss more in the next section. Briefly, our approach for optimal PMU placement

using uncertainty evaluation can be described by Algorithm 2.

2.8.3 Simulation Results

In this section, we first apply our observability checking and estimation uncertainty analysis

to two simulated multi-machine systems. We then apply our approach to evaluate four different

optimal PMU placement alternatives, and find the best one.

32

Algorithm 2: Optimal PMU Placement for Static State EstimationBuild the optimization problem as shown in (2.64) based on the network topology;if there is more than one solution to this problem then

for each solution set of PMUs doEvaluate A according to section 2.8.1;Evaluate Q according to section 2.8.1;Evaluate H according to section 2.8.1;Evaluate R according to section 2.8.1;Compute Ψ using (2.11);Compute the n eigenvectors of Ψ using (2.12);Compute P∞ using (2.13);

Apply the uncertainty values to user-defined criterion;Determine the optimal PMU placement strategy;

elseReturn the only possible solution set of PMUs;

Observablity Checking and Estimation Uncertainty

Given the sets of PMU buses and zero injection buses, Algorithm 1 checks observability

by returning a list of observable buses classified at different levels. Then the steady state error

covariance P∞ of these observable buses is computed, with each diagonal element P∞ii (i.e.

the variance) indicating the estimation uncertainty of the corresponding bus. The estimation

uncertainty of any unobservable bus i is P∞ii = ∞. To better illustrate the results, we define

1/√P∞ii to be the estimation “certainty” (information) of the corresponding bus. In this way,

the certainty of any unobservable bus is zero; as for any observable bus, the higher certainty

value it has, the better it can be estimated.

We simulated our algorithm on one small system consisting of three machines in a looped

network of nine buses as shown in Fig. 2.2.

33

Figure 2.2: The 3-machine 9-bus system

Table 2.1 presents the observability analysis of three simulated cases. Fig. 2.3, 2.4 and

2.5 illustrates the certainty plots respectively. In all the following certainty plots, the green-

colored numbers in cirles represent the PMU buses, while the blue-colored numbers in squares

represent the zero-injection buses.

Table 2.1: Observability Checking for the 3-machine 9-bus System

Case PMU buses 0-inj buses Observable buses

1 1, 2 None L1: 1, 2

L2: 4, 8

2 1, 2, 3 None L1: 1, 2, 3

L2: 4, 6, 8

3 1, 2, 3 9 L1: 1, 2, 3

L2: 4, 6, 8

L3: 9

34

0

50

100

150

3

6

7

2

Steady State Estimation Certainty

8

5

94

1

Cer

tain

ty(v

olt−

1)

Figure 2.3: The certainty analysis plot of case1 from Table 2.1, in the 3-machine 9-bus sys-tem

0

50

100

150

3

6

7

2


8

5

94

1

Cer

tain

ty(v

olt−

1)


0

50

100

150

3

6

7

2


8

5

94

1

Cer

tain

ty(v

olt−

1)


Then similarly, we simulated our method on a larger 16-machine 68-bus system as shown

in Fig. 2.6, representing the interconnected New England Test System and New York Power

System (NETS-NYPS) [24], with also three cases shown in Table 2.2 and the certainty analysis

35

of each one in Fig. 2.7, 2.8 and 2.9.

Table 2.2: Observability Checking for the 16-machine 68-bus System

Case PMU buses 0-inj buses Observable buses

1 1, 11, 21, None L1: 1, 11, 21

L2: 2, 6, 10, 12, 16, 22,

27, 30, 31, 47

2 1, 11, 21, None L1: 1, 11, 21, 31, 41, 51

31, 41, 51 L2: 2, 6, 10, 12, 16, 22,

27, 30, 38, 40, 42, 45,

47, 50, 62, 66

3 1, 11, 21, 12, 38, 40, L1: 1, 11, 21, 31, 41, 51

31, 41, 51 42, 50 L2: 2, 6, 10, 12, 16, 22,

27, 30, 38, 40, 42, 45,

47, 50, 62, 66

L3: 13 48 52

L4: 67

Figure 2.6: The 16-machine 68-bus system

36

0

20

40

60

80

100

120

140

160

180

5923

58

55

5756

22

20

65

10

19

37

21

13

7

6124

5411

43

16

64

29

12

39

14

44

15

36

6

26

8

28

5

1727

4

18

45


35

25

34

60

68

9

50

3

52

51

33

2

32301

53

6362

47

49

3831

67

48

42

46

40

4166

Figure 2.7: The certainty analysis plot of case 1 from Table 2.2, in the 16-machine 68-bussystem

0

50

100

150

200

5923

58

55

5756

22

20

65

10

19

37

21

13

7

6124

5411

43

16

64

29

12

39

14

44

15

36

6

26

8

28

5

1727

4

18

45


35

25

34

60

68

9

50

3

52

51

33

2

32301

53

6362

47

49

3831

67

48

42

46

40

4166


37

0

50

100

150

200

5923

58

55

5756

22

20

65

10

19

37

21

13

7

6124

5411

43

16

64

29

12

39

14

44

15

36

6

26

8

28

5

1727

4

18

45


35

25

34

60

68

9

50

3

52

51

33

2

32301

53

6362

47

49

3831

67

48

42

46

40

4166


Comparison Among Multiple Optimal Solutions

In this subsection, we used the small system in Fig. 2.2 as an example and only consid-

ered PMU measurements. By solving the traditional optimization problem, we could in fact

obtain four “optimal” solutions, with PMUs installed at buses 1, 6, 8, 2, 4, 6, 3, 4, 8 and

4, 6, 8 respectively. They are all “optimal” in the sense that they all make the entire sys-

tem observable by using a minimum number of PMUs. To determine which PMU placement

strategy is the best, we demonstrate the certainty plots of them in Fig. 2.10-2.13.

38

0

50

100

150

3

6

7

2


8

5

94

1

Cer

tain

ty(v

olt−

1)

Figure 2.10: PMUs installed at buses 1, 6 and8

0

50

100

150

3

6

7

2


8

5

94

1

Cer

tain

ty(v

olt−

1)


0

50

100

150

3

6

7

2


8

5

94

1

Cer

tain

ty(v

olt−

1)


0

50

100

150

3

6

7

2


8

5

94

1

Cer

tain

ty(v

olt−

1)


Furthermore, it is up to the users to decide what criterion they prefer to use in making

their specific choices. For instance, one can use the maximum, the minimum, the mean of

these “certainties” (Fig. 2.14), or even some weighted function of them. From a practical point

39

of view, the users may want to choose the minimum as their criterion, because for the power

system to function properly, even the worst estimates should exceed a certain standard. In our

example, we can immediately tell that by all means, the PMU placement at buses 4, 6, 8

outperforms other strategies.

Figure 2.14: A comparison of steady-state certainties for the four “optimal" solutions, basedupon three criteria: maximum (max), minimum (min) and average (mean).

2.9 My Method: PMU Placement for Power System Dynamic State Esti-

mation

2.9.1 PMU Placement Evaluation

Previously we presented a stochastic approach to characterize the observability and corre-

sponding uncertainty of the power system buses for any given configuration of PMUs, achiev-

ing full observability or not. Here we connect the optimal PMU placement with power system

40

dynamic state estimation: we restate the optimal PMU placement problem such that the solu-

tion is not only a minimum set of PMUs that can cover the entire power system, but also the

one which benefits dynamic state estimation the most.

The Process Model

Without loss of generality, in a power system that consists of n generators, let us consider

the generator i which is connected to the generator terminal bus i. We use a classical model

for the generator composed of a voltage source |Ei|∠δi with constant amplitude behind an

impedance X ′di

. As given in [25], the non-linear differential-algebraic equations regarding the

generator i can be written asdδidt

= ωB(ωi − ω0)

dωi

dt= ω0

2Hi(Pmi

− |Ei||Vi|X′

di

sin(δi − θi)−Di(ωi − ω0))(2.67)

where state variables δ and ω are the generator rotor angle and speed respectively, ωB and ω0

are the speed base and the synchronous speed in per unit 2, Pmiis the mechanical input, Hi is

the machine inertia3, Di is the generator damping coefficient and |Vi|∠θi is the phasor voltage

at the generator terminal bus i (which is a function of δ1, δ2, ..., δn).

For the state vector x = [δ1, ω1, δ2, ω2, ..., δn, ωn]T , the corresponding continuous time

change in state can be modeled by the linearized equation

dx

dt= Acx+ wc, (2.68)

where wc is an 2n × 1 continuous time process noise vector with 2n × 2n noise covariance

matrix Qc = E[wcwTc ], and Ac is an 2n× 2n continuous time state transition Jacobian matrix,

2In the power systems analysis field of electrical engineering, a per-unit system is the expression of sys-tem quantities as fractions of a defined base unit quantity. Readers can refer to http://en.wikipedia.org/wiki/Per-unit_system

3The mechanical power Pmi and machine inertia Hi should not be confused with the error covariance P andmeasurement Jacobian H from the preceding section. While potentially confusing these are the variables used bypopular convention in the respective fields.

41

whose entries for i ∈ 1, ..., n and j ∈ 1, ..., n (i = j) are the corresponding partial

derivatives [25]

Ac[2i−1,2i−1]= 0 (2.69)

Ac[2i−1,2i]= ωB (2.70)

Ac[2i,2i−1]= − ω0|Ei|

2HiX ′di

[∂|Vi|∂δi

sin(δi − θi) (2.71)

+|Vi| cos(δi − θi)(1−∂θi∂δi

)

]Ac[2i,2i] = −

ω0

2Hi

Di (2.72)

Ac[2i−1,2j−1]= 0 (2.73)

Ac[2i−1,2j]= 0 (2.74)

Ac[2i,2j−1]= − ω0|Ei|

2HiX ′di

[∂|Vi|∂δj

sin(δi − θi) (2.75)

+|Vi| cos(δi − θi)(−∂θi∂δj

)

]Ac[2i,2j] = 0. (2.76)

Hence the update of the state vector x from time step (k − 1) to k over duration ∆t has the

complete corresponding discrete-time state transition matrix

A = I + Ac ·∆t. (2.77)

The next issue is the discrete-time process noise covariance Q. As described by [20], if we

assume the process noise “flows" through (is shaped by) the same system of integrators repre-

sented by A, we can integrate the continuous time process equation (3.48) over the time interval

∆t to obtain a 2n× 2n discrete-time (sampled) Q matrix:

Q =

∫ ∆t

0

eActQceAT

c tdt (2.78)

Now written in the form of the process model from equation (2.1) we have

xk = Axk−1 + wk−1 (2.79)

= (I + Ac ·∆t)xk−1 + wk−1, (2.80)

42

where w is the process noise with normal probability distribution p(w) ∼ N(0, Q).


For now we only consider the measurements provided by PMUs. According to [5–7], and

our analysis of the bus observation “directness" in Section 2.8.1, all the observable buses are

divided into two levels: Level 1 contains the directly observable buses with PMUs installed

on them; while Level 2 contains the more “indirect” observable buses with PMUs installed on

their neighbors instead of themselves.

As a result, the measurements are functions of the phasor voltages of all observable buses:zV

zC

=

I 0

YCL1 YCL2

VL1

VL2

+

vV

vC

(2.81)

We can simply denote the equation above as

z′ = H ′′V + v′ (2.82)

However in this section, our state variables are no longer bus voltages, but dynamic gener-

ator state variables, so we need to introduce the expanded system nodal equation first [11]:

Yexp

E

V

=

YGG YGL

YLG YLL

E

V

=

IG

0

(2.83)

where E is the vector of internal generator complex voltages, V is the vector of bus complex

voltages, IG represents electrical currents injected by generators, Yexp is called the expanded

nodal matrix, which includes loads and generator internal impedances: YGG, YGL, YLG and YLL

are the corresponding partitions of the expanded admittance matrix.

Therefore the relationship of the bus voltages to the generator voltages can be expressed as

[11]:

V = −Y −1LL YLGE = RVE (2.84)

43

where RV is defined as the bus reconstruction matrix.

Combining equations (2.82) and (2.84), we have:

z′ = H ′′RVE + v′ = H ′E + v′ (2.85)

Because the internal generator complex voltage vector E = [|E1|∠δ1, |E2|∠δ2, ..., |En|∠δn]T

is a function of the state vector x, the measurements can be modeled as

z = h(x) + v (2.86)

where z = [|z′1|, arg(z′1), |z′2|, arg(z′2), ..., |z′m|, arg(z′m)]T is the 2m × 1 measurement vector

that consists of amplitude and phase angle of each PMU measurement.

Furthermore, the measurement model can be expressed in the form of equation (2.2) after

linearization

z = Hx+ v (2.87)

where H is the corresponding Jacobian matrix [25]:

H =

∂h1

∂δ10 ∂h1

∂δ20 · · ·

∂h2

∂δ10 ∂h2

∂δ20 · · ·

∂h3

∂δ10 ∂h3

∂δ20 · · ·

∂h4

∂δ10 ∂h4

∂δ20 · · ·

......

...... . . .

, (2.88)

and v is the measurement noise with normal probability distribution p(v) ∼ N(0, R). The

measurement noise covariance R can be determined experimentally, by testing the PMUs over

time.

44

2.9.2 Optimal PMU Placement

Here the PMU placement evaluation using steady-state uncertainty analysis is very similar

to the method described in section 2.8.2, except that we are dealing with dynamic states now.

In order to compute P∞ for all desired points in the state space, we need to identify the pa-

rameter matrices A, Q, H and R. Note that in these matrices, only the generator rotor angles

δ1, δ2, ..., δn are variables, so one only need to decide at which interest points x1, x2, ...xp

where xi ∈ [0, 2π]n, to evaluate P∞. Finally, our approach is described by Algorithm 3.

Algorithm 3: Optimal PMU Placement for Dynamic State EstimationBuild the optimization problem as shown in (2.64) based on the network topology;if there is more than one solution to this problem then

for each solution set of PMUs dofor each n-dimensional point xi ∈ x1, x2, ...xp do

Determine ∆t for the PMUs;Evaluate A with ∆t using (2.77);Evaluate Q with ∆t using (2.78);Evaluate H using (2.88);Evaluate R;Compute Ψ using (2.11);Compute the n eigenvectors of Ψ using (2.12);Compute P∞

i using (2.13);Compute an aggregated uncertainty value from P∞

i

Apply the aggregated uncertainty values to user-defined criterion;Determine the optimal PMU placement strategy;

elseReturn the only possible solution set of PMUs;


In this section, we first simulate the uncertainty analysis on an ideal multi-machine system

with PMU installed on every bus. Then we applied our method to evaluate different optimal

PMU placement strategies and find the best solution.

45

Estimation Uncertainty of a Fully PMU-installed System

We carried out our steady-state uncertainty analysis on the same small test system consist-

ing of three machines in a looped network of nine buses as shown in Fig. 2.2. The dynamic

state vector is x = δ1, ω1, δ2, ω2, δ3, ω3. To better illustrate the results, we represent the entire

set of interest points x = δ1, δ2, δ3 ∈ [0, 2π]3 by a 3D grid. For each point xi, we obtain a

6 × 6 steady-state error covariance matrix P∞i , and then aggregate the information in P∞

i to

the following value for visualization

f(xi) =

∑j=1,3,5

√P∞i[j,j]

+ wB

∑j=2,4,6

√P∞i[j,j]·∆t

3, (2.89)

f(xi) =

∑k=1,2,3

(√

P∞i[2k−1,2k−1]

+ wB

√P∞i[2k,2k]

·∆t)

3(2.90)

where we use ∆t to convert speeds to changes in angle over the inter-measurement period.

Because

√P∞i[2k−1,2k−1]

+ wB

√P∞i[2k,2k]

·∆t (2.91)

represents the uncertainty in estimating the rotor angle of generator k over time ∆t, the value

f(xi) is the average uncertainty of the three generators over ∆t at this particular point xi.

In this extreme case, we assume all buses are equipped with PMUs. Fig. 2.15 illustrate

the f(x) values throughout the [0, 2π]3 space. One can clearly tell a periodic pattern in the

plot, because the Jacobian parameter matrices involve trigonometric functions of x. The darker

areas reflect lower uncertainty, and imply better expected estimation. This ideal simulation

indicates the best estimation uncertainty level we can reach with a PMU placed on every bus.

46

Figure 2.15: Steady-state estimation uncertainty versus rotor angles for the ideal case, i.e.when PMUs are installed on all buses.

Comparison Among Multiple Optimal Solutions

We used the same system in Fig. 2.2 as an example of finding the optimal PMU placement

solution. After solving the optimization problem (2.64) with the linear integer programming

method, we obtained four solutions, with PMUs installed at buses 1, 6, 8, 2, 4, 6, 3, 4, 8

and 4, 6, 8 respectively. According to previous research they can all be called “optimal” in

the sense that they each make the entire system observable by using a minimum number of

PMUs. To visualize the differences between these four PMU placement strategies via our ap-

proach, we provide the uncertainty plots for them in Fig. 2.16–2.19 using the same visualization

method in Fig. 2.15, where darker color depicts lower uncertainty value.4

4Each of Fig. 2.16–2.19 uses the same colormap and plot range.

47

Figure 2.16: Steady-state estimation uncer-tainty versus rotor angles for PMUs installed atbuses 1, 6 and 8.




Apparently, Fig. 2.19 has darker colors corresponding to better performance, as compared

to Fig. 2.16–2.18. See also Fig. 2.20. One can choose different criteria depending on the cir-

cumstances. For instance, one could use the maximum, minimum, or mean of these aggregated

uncertainties as depicted in Fig. 2.20, or even some weighted function of them. In our example,

the scheme with PMUs placed at buses 4, 6, 8 is more likely to be chosen by the designer,

48

because it has smaller max, min and mean uncertainties than the alternative scenarios. In prac-

tice, many factors may affect the final decision, e.g., existing infrastructure, cost, etc. Our

quantitative approach helps guide decision-making by providing concrete information about

tradeoffs, which can not be offered by pure network topology analysis.

Figure 2.20: The comparison of the four “optimal" solutions based upon three criteria: max,min and mean

49

CHAPTER 3

FILTER COMPUTATION ADAPTATION

As measuring devices continue to develop towards higher accuracy and update frequency,

the amount of measurement data associated with large complex systems becomes enormous.

For example, in power systems that are usually large-scale and wide-area interconnected, the

computational requirement for data processing has made real-time estimation a real challenge.

In this chapter we present a new method called Lower Dimensional Measurement-space

(LoDiM) state estimation. It is an approach featuring a dynamic measurement selection pro-

cedure: a small measurement subset is dynamically and strategically chosen for each cycle to

process. The smaller measurement-space yields smaller computational cost and hence lower

reporting latency. In specific for power system dynamic state estimation, we present the Re-

duced Measurement-space Dynamic State Estimation (ReMeDySE) derived from LoDiM.

Although LoDiM is presented in the context of the Kalman filter, the associated measure-

ment selection procedure is not filter-specific, i.e. it can be used with other state estimation

methods such as particle and unscented filters. LoDiM, along with modern parallel compu-

tation techniques, can facilitate other large-scale, real-time and computation-intensive state

tracking systems beyond the power systems.

3.1 Introduction

State estimation plays a basic and important role in modern industries. Particularly for

power systems, it seeks to produce a reliable dynamic database for use in critical operational

functions including real-time security monitoring, load-forecasting, economic despatch, and

50

load-frequency control.

There are typically four steps involved in power system state estimation procedures [21]:

1. Modeling. In this step, network topology is determined, and preliminary data is checked

and calculated.

2. Observability determination. Various sensor measurements are classified as either crit-

ical or redundant. Critical measurements are necessary to achieve the system observ-

ability. Therefore, if some critical measurements are lost, pseudo measurements can be

included. On the other hand, redundant measurements can be removed without affecting

the system observability. A redundant measurement may belong to some critical pairs or

critical k-tuples. A critical pair stands for two redundant measurements whose simulta-

neous removal from the measurement set will make the system unobservable. Similarly,

a critical k-tuple contains a set of k redundant measurements, whose simultaneous re-

moval will cause the system to become unobservable.

3. State estimation. Typically, a state estimator receives telemetered measurements from

SCADA system in the time interval of several seconds. From the measurements and

an assumed steady state system model, the state estimator generates a set of static state

variables as a best estimate of the system conditions.

4. Detection and identification of gross measurement errors and/or errors in network struc-

ture.

Traditional SCADA systems collect RTU measurements. Updated every 2 to 4 seconds,

they are too infrequent to capture the system dynamics, e.g. short circuits, line switching, load

changes, etc. There has not been an issue historically, because traditional state estimators up-

date the estimates every 3 to 5 minutes. Under the steady state system model assumption, none

of the dynamic characteristics of a power system are reflected in the estimation. However, this

changed with the introduction of modern phasor measurement technologies. High-speed and

51

high-accuracy synchrophasor measurements, combined with dynamic estimation techniques

such as the Kalman filters, have motivated real-time power system state estimation.

Due to the tremendously increasing size and complexity of the interconnected power net-

works, tedious computations in real-time state estimation remain obstacles to overcome. Even

with modern supercomputers, the massive data processing is still a very challenging task given

the time and space (memory) complexities, despite significant work has been done aiming at

accelerating the calculations via parallel implementations, such as OpenMP [26] and Global

Arrays [27].

Here we introduce a different approach to reduce the computational burden. Previously we

introduced an estimation approach called single-constraint-at-a-time (SCAAT) [14]. The idea

behind the SCAAT approach is to use a Kalman filter to estimate a globally observable system

using low dimensional measurements from potentially unobservable subspaces. In each filter

cycle, SCAAT deals with much lower dimensional measurement data. The benefits include

higher sampling rates, lower latencies, and improved accuracy. Although SCAAT was first

developed for optoelectronic 3D tracking systems, it inspired us to design LoDiM [15, 16]

and ReMedySE [17]. They are SCAAT-based algorithms for large-scale system state tracking,

featuring dynamic measurement selection to reduce the computational burden.

In this Chapter, we first present a comprehensive study and proof of LoDiM, including the

trade-offs and optimal number of measurements to be selected; and then we present ReMe-

DySE, which is derived and expanded from LoDiM to capture the dynamic states in power

systems. Our method can significantly facilitate power system operations, and integrate with

the powerful tool of hierarchical, distributed estimation as well as parallel computation tech-

niques for more improvement.

52


The remainder of this chapter is organized as follows. Section 3.3 introduces the back-

ground. In Section 3.4, we review the traditional Kalman filter techniques, including Kalman

filter (KF) and Extended Kalman filter (EKF). Section 3.5 gives a brief description of the

SCAAT approach, which is an incremental tracking method with incomplete information. Sec-

tion 3.6 presents our new approach, Lower Dimensional Measurement-space (LoDiM) state

estimation, which performs Kalman filter upon dynamically selected measurements. It reduces

the computation requirement dramatically, without sacrificing the tracking quality. We simu-

lated the method on two power system models in different sizes to demonstrate and analyze the

performance. Furthermore, we expand our approach to power system dynamic state estimation

model, and developed Reduced Measurement-space Dynamic State Estimation (ReMeDySE)

in Section 3.7, particularly for power system dynamic state estimation.


In modern power systems, measurement devices with higher accuracy and update rate are

increasingly deployed, setting new requirements for real-time computations. Significant pre-

vious work has attempted to alleviate the computational load problem. One approach is to

explore the potential computational power: the authors of [28] used Petri net (PN) theory to

achieve the optimum utilization of processors, in a state estimator based on Kalman filter.

Another attempt is to reduce the computational complexity. For example, in [29] the au-

thors proposed a method for systems that have more measurements than states. They described

an equivalent state-space system in which the number of measurements equals the number of

states. The system model is ideally assumed static and linear, and the preparation overhead

was ignored in the complexity analysis.

53

[30] illustrated a measurement selection procedure using an Extended Kalman filter. As

stated by the authors, an inherent limitation of the proposed method is that measurement selec-

tion is based entirely on the steady-state sensitivity matrix. According to this approach: (1) the

actual information content of the candidate measurements under typical operating conditions is

not considered; (2) the measurement rankings obtained are local and dependent on the steady

state chosen as the base case; and (3) dynamic and non-linear effects are not considered.

[14] presented SCAAT, a Kalman-filter-based incremental tracking algorithm using incom-

plete information. It estimates a globally observable system using only measurements from

locally unobservable systems. The underlying principle is that the single observations provide

some information about the user’s state, and thus can be used to incrementally improve a pre-

vious estimate. Based on SCAAT, another category called MCAAT, which processes multiple

constraints at a time, was further derived.

This inspired us to propose LoDiM with sophisticated measurement selection procedure

[15, 17]. It is also a Kalman-filter-based approach, with higher reporting rates and lower la-

tency. Then we take the dynamic states of power systems into consideration and present ReM-

eDySE [17], an estimator that is suitable for reflecting power system dynamic characteristics.

In [31] we considered another approach to reduce the computational load of an ensemble

Kalman filter. The main idea is to identify and process the most informative measurement

subspace based on their capability to reduce the trace of the a posteriori error covariance. We

construct a low-dimensional measurement set from linear combinations of the original mea-

surement set, by computing a transformation matrix during each cycle. When the number of

measurements is large and number of non-zero covariance eigenvalues is small, this algorithm

can make an effective trade-off between computational complexity and estimation efficacy.

54

3.4 The Kalman Filter Techniques

Because of their efficiency, robustness and typical accuracy, Kalman filters [13, 19] have

been employed in a wide range of applications from economic analysis to space navigation, and

are especially popular in the areas of motion tracking systems. In power system applications,

they have been extensively used to improve the computational performance of traditional state

estimation process since the 1970’s [32, 33]. Recent study [11, 34] also applied them to power

system dynamic state models.

Since here we focus on discrete Kalman filters, we will omit the word “discrete" for the

rest of this thesis. The Kalman Filter is a stochastic estimator for the instantaneous state of a

dynamic system. The filter is very powerful in several aspects: it supports estimations of past,

present, and even future states, and it can do so even when the precise nature of the modeled

system is unknown. The Extended Kalman Filter was proposed later for non-linear models. In

this section, we will give a brief overview of the discrete Kalman Filter (KF) and the Extended

Kalman Filter (EKF). Readers could refer to [35] for a more detailed introduction.

3.4.1 The Kalman Filter

The Kalman filter, or KF, is a set of mathematical equations that provides an efficient com-

putational (recursive) means to estimate the state of a process, in a way that minimizes the mean

of the squared error. In order to apply the Kalman filter, we must first have a hypothetically

true model of the system in hand.

Recall that in Section 2.4, we have introduced a generalized linear system model:

xk = Axk−1 +Buk−1 + wk−1 (3.1)

zk = Hxk + vk (3.2)

55

The Kalman Filter estimates the state by minimizing the a posteriori estimate error covari-

ance, in a recursive prediction-correction manner. The prediction step is realized by a set of

time update equations:

Prediction Step

1. Project the state ahead:

x−k = Axk−1 +Buk−1 (3.3)

We define x−k ∈ Rn to be the a priori state estimate at time step k given the knowledge

of the process prior to k, in this case, the a posteriori state estimate xk−1 at time step

k − 1, the state transition matrix A, and the control input (optional).

2. Project the error covariance ahead:

P−k = APk−1A

T +Q (3.4)

We define e−k ≡ xk − x−k to be the a priori estimate error, so P−

k ≡ E[e−k e−Tk ] is the a

priori estimate error covariance, where E denotes statistical expectation.

The time update equations are responsible for projecting forward (in time) the previous

state xk−1 and error covariance estimates Pk−1 to obtain the a priori estimates for the next time

step k.

The correction step is carried out by a set of measurement update equations:

Correction Step

1. Compute the Kalman gain:

Kk = P−k HT (HP−

k HT +R)−1 (3.5)

Where K is a n×m matrix called the Kalman gain matrix,

56

2. Update the error covariance:

Pk = (I −KkH)P−k (3.6)

ek ≡ xk − xk is called the a posteriori estimate error, and Pk ≡ E[ekeTk ] becomes the a

posteriori estimate error covariance, where E denotes statistical expectation. Pk contains

important information reflecting the filter’s uncertainty in the estimation.

Under certain conditions when the system is controllable and observable, as k →∞, the

limiting behavior of the Kalman filter can result in a bounded steady-state error covari-

ance P [36], if we have

limk→∞

P−k = lim

k→∞Pk = P <∞ (3.7)

which means if the time is long enough, more iterations of the Kalmn filter will no longer

affect the value of error covariance P , whether a priori or a posteriori. Thus by omitting

the subscript “k" and superscript “+" [10], we have:

P = (I −KH)P (According to equation (3.6))

= P −KHP

= P − PHT (HPHT +R)−1HP (According to equation (3.5))

= A(P − PHT (HPHT +R)−1HP )AT +Q (According to equation (3.4))

= APAT − APHT (HPHT +R)−1HPAT +Q

This error covariance expression is the Discrete Algebraic Riccati Equation (DARE), as

seen in equation (2.10) from Section 2.5.

3. Update state estimate with measurement zk:

xk = x−k +Kk(zk −Hx−

k ) (3.8)

We define xk ∈ Rn to be the a posteriori state estimate at time step k.

57

zk is the actual measurement at time step k, and Hx−k is the predicted measurement at

time step k, The difference (zk −Hx−k ) is conventionally called the innovation.

K reflects how we trust the actual measurement zk versus the predicted measurement

Hx−k . From its expression, one can tell that larger values of R place more weight on the

predicted value while smaller values of R place more weight on the measured values.

There is an alternative formulation for equations (3.5) and (3.6) in correction steps 1 and 2,

they are:

1′ Update the error covariance:

Pk = [(P−k )−1 +HTR−1H]−1 (3.9)

Equation (3.9) is equivalent to (3.6) [36], because according to equation (3.6),

(Pk)−1 = ((I −KkH)P−

k )−1

= (P−k )−1(I −KkH)−1

= (P−k )−1(I − P−

k HT (HP−k HT +R)−1H)−1

= (P−k )−1(H−1(HP−

k HT +R)(HP−k HT +R)−1H − P−

k HT (HP−k HT +R)−1H)−1

= (P−k )−1((P−

k HT +H−1R− P−k HT )(HP−

k HT +R)−1H)−1

= (P−k )−1((H−1R)(HP−

k HT +R)−1H)−1

= (P−k )−1(H−1(HP−

k HT +R)R−1H)

= (P−k )−1(P−

k HTR−1H + I)

= (P−k )−1 +HTR−1H

By applying the inverse operations on both sides, we have

Pk = [(P−k )−1 +HTR−1H]−1

58

2′ Compute the Kalman gain:

Kk = PkHTR−1 (3.10)

The equivalence of two Kalman gain formulations in equations (3.10) and (3.5) can be

proved as follows [36]:

Kk = P−k HT (HP−

k HT +R)−1

= PkP−1k P−

k HT (HP−k HT +R)−1

= Pk((P−k )−1 +HTR−1H)P−

k HT (HP−k HT +R)−1 (According to equation (3.9))

= Pk(HT +HTR−1HP−

k HT )(HP−k HT +R)−1

= PkHTR−1(R +HP−

k HT )(R +HP−k HT )−1

= PkHTR−1

One advantage of equations (3.5) and (3.6) over equations (3.9) and (3.10) is that only

one matrix inversion is required, and we do not need to worry about the singularity of

matrix P−k , that is, whether |P−

k | = 0.

The measurement update equations are responsible for the feedback, i.e. for incorporating

a new measurement into the a priori estimate to obtain an improved a posteriori estimate.

To sum up, the time update (“predictor") projects the current state estimate ahead in time,

and the measurement update (“corrector") adjusts the projected estimate by an actual measure-

ment at that time. We can visually express the final estimation algorithm by a Kalman filter

cycle as shown in Fig. 3.1: once the initial estimates for x0 and P0 are fed into the filter, it

evolves the state estimates in a recursive predictor-corrector fashion.

59

Time Update

“Predict”

Measurement Update

“Correct”

Initial Estimates

Figure 3.1: The operation of a discrete Kalman filter cycle

3.4.2 The Extended Kalman Filter

In practice, the process model and (or) the measurement relationship to the process is usu-

ally non-linear. A non-linear system model can be generalized as:

xk = a(xk−1, uk−1, wk−1) (3.11)

zk = h(xk, vk). (3.12)

For instance, we can only express the power system dynamic states variables in non-linear

differential-algebraic process and measurement equations, representing the power system dy-

namic behavior:

dx

dt= f(x, y) (3.13)

zk = h(xk, vk)

60

In generalized non-linear system form, the equations become:

xk = xk−1 + f(xk−1, yk−1)∆t (3.14)

zk = h(xk, vk)

Our objective is to estimate these dynamic states, and EKF was proposed to deal with such

problems.

We could approximate the state and measurement vectors in (3.11) and (3.12) by:

xk = a(xk−1, uk−1, 0) (3.15)

zk = h(xk, 0) (3.16)

where xk−1 is an a posteriori estimate of the state at step k − 1.

Then by linearizing the estimation about xk and xk, we obtain a new set of linear difference

equations and measurement equation for the error process:

(xk − xk) ≈ Ak(xk−1 − xk−1) +Wkwk−1 (3.17)

(zk − zk) ≈ Hk(xk − xk) + Vkvk (3.18)

where

xk and zk are the true state and measurement vectors at step k,

xk and zk are the approximated state and measurement vectors at step k.

xk−1 is an a posteriori estimate of the state vector at step k − 1.

Ak =∂a∂x|xk−1,uk−1,0 is the Jacobian matrix of partial derivatives of a with respect to x,

Wk =∂a∂w|xk−1,uk−1,0 is the Jacobian matrix of partial derivatives of a with respect to w,

Hk =∂h∂x|xk,0 is the Jacobian matrix of partial derivatives of h with respect to x,

61

Vk =∂h∂v|xk,0 is the Jacobian matrix of partial derivatives of h with respect to v.

The Jacobian matrices A, W , H and V have time step subscript k because they may vary

at each time step.

For simplicity, we denote the prediction error as

exk= xk − xk, (3.19)

and measurement residual as

ezk = zk − zk. (3.20)

We define

εk = Wkwk−1 (3.21)

ηk = Vkvk (3.22)

as two new mutually independent random variables with normal probability distributions:

p(εk) ∼ N(0,WkQk−1WTk ) (3.23)

p(ηk) ∼ N(0, VkRkVTk ) (3.24)

Thus we write the error process 3.17 as

exk≈ Ak(xk−1 − xk−1) + εk (3.25)

ezk ≈ Hk(xk − xk) + ηk (3.26)

for short. Note that this is a linearized system for us to estimate conveniently by applying

Kalman Filter.

An Extended Kalman filter, or EKF, is nothing but a Kalman filter that linearized about the

current mean and covariance. With some slight modifications to the KF equations, the EKF

62

has the following steps:

Prediction :

x−k = a(xk−1, uk−1, 0)

P−k = AkPk−1A

Tk +WkQk−1W

Tk

(3.27)

Correction :

Kk = P−

k HTk (HkP

−k HT

k + VkRkVTk )−1

xk = x−k +Kk(zk − h(x−

k , 0))

Pk = (I −KkHk)P−k

(3.28)

3.5 The SCAAT Method

[37] has argued that there is a direct relationship between the complexity of the estimation

algorithm, the corresponding speed (execution time per estimation cycle), and the change in

system state between estimation cycles.

This relationship can be clearly visualized with the Complexity-Speed-Change cycle in

Fig. 3.2 [1]. The notes on the outside of the circle represent a vicious cycle: as the algorithmic

complexity increases, the corresponding execution time increases, which allows for significant

(and usually non-linear) system state changes between estimation cycles. This in turn implies

the need for a more complex dynamic process model and estimation algorithm.

The single-constraint-at-a-time (SCAAT) approach, first proposed and described in [14],

is a Kalman-filter-based incremental tracking algorithm with an attempt to reverse this cycle.

Instead of waiting to form and process a complete collection of measurements (constraints)

like the traditional Kalman filter would do, the SCAAT method intentionally fuses a single

measurement per estimate cycle. Indeed, single measurements underconstrain the mathemati-

cal solution, however, they still can provide some information about the system state, and thus

incrementally improve a previous estimate. By applying SCAAT, the algorithmic complexity

is drastically reduced, which reduces the corresponding execution time, and hence reduces the

amount of state changes between estimation cycles. To deal with limited state changes, we

63

only need to use a simple dynamic process model in the estimation algorithm, which further

simplifies the computations. The SCAAT has formed a cycle as denoted on the inside of the

circle.

HIGH

HIGH

LESS

LESS

LOWLOW

MORE

MORE

LOW

LOW

HIGH

HIGH

Complexity

Figure 3.2: Complexity-Speed-Change Cycle [1]

3.6 My Method: Lower Dimensional Measurement-space (LoDiM)State Estimation

In a common scenario where the processor running the estimation algorithm needs to exe-

cute some additional tasks (e.g., simulation of an urgent contingency), we will have to release

some computational resources for the tasks. Ideally, we want the estimation process to con-

tinue with reduced computational cost, but without sacrificing the reporting rate. LoDiM is

thus proposed for this purpose.

The LoDiM method employs a Kalman filter that incorporates lower dimensional (sub-

space) measurements in each cycle, to adapt to limited computational resources. It employs

a special measurement selection procedure to strategically reduce the measurement-space di-

64

mension. As a result, it yields a much smaller computational load, lower latency, and most

importantly, reliable performance.

In this section, we will first give a brief description of the design principle behind LoDiM,

and then discuss our dynamic measurement selection method and the structure of LoDiM.

3.6.1 Principle of Design

As one would expect, performing dynamic state estimation with KF/EKF (or any filter)

is a rather computationally intensive process. For small systems, the computation may be fast

enough for real time control applications; However the growth of three factors will significantly

affect the computational effort: the size of the system, the complexity of model components,

and the number of measurements to be processed. Table 3.1 [38] has listed the upper bound

on computational requirements of the Kalman filter, measured by FLoating-point OPeration

(FLOP) counts. FLOP takes into account matrix multiplications and additions. Computational

cost in the table is considered upper bound because the implementation steps are not yet subject

to various optimizations, such as storage saving, data reuse, etc. More detailed discussions can

be found in [38, 39].

65

Table 3.1: Kalman Filter Computational Cost Upper Bound

Step Variable Computation Breakdown FLOP Breakdown FLOP counts

1 x−k Axk−1 2n2 − n 2n2 − n

2 P−k Pk−1A

T 2n3 − n2 4n3 − n2

APk−1AT 2n3 − n2

APk−1AT +Q n2

3 Kk HP−k 2n2m− nm 2m3 + 4nm2 + 2n2m− 2nm

HP−k HT 2nm2 −m2

HP−k HT +R m2

(HP−k HT +R)−1 2m3

(P−k HT )(HP−

k HT +R)−1 2nm2 − nm

4 xk Hx−k 2nm−m 4nm

zk −Hx−k m

Kk(zk −Hx−k ) 2nm− n

x−k +Kk(zk −Hx−

k ) n

5 Pk Kk(HP−k ) 2nm2 −m2 n2 + 2nm2 −m2

P−k −Kk(HP−

k ) n2

Adding up the FLOP counts at each step in Table 3.1 yields an estimate of the computational

cost for a Kalman filter cycle, which is at most:

(2n2 − n) + (4n3 − n2) + (2m3 + 4nm2 + 2n2m− 2nm) + (4nm) + (n2 + 2nm2 −m2)

= 2m3 + 6nm2 + 2n2m+ 4n3 + 2n2 + 2nm−m2 − n (3.29)

We have noticed the expensive cost of calculating the Kalman gain Kk in step 3, because

it involves the inversion of a m×m matrix (HP−k HT + R), with complexity of O(m3). This

makes the computation intractable when the number of measurements m is too large for the

available computational resource to handle promptly, which unfortunately, can happen at any

moment.

On the other hand, if we can reduce the measurement-space dimension to adapt to limited

66

computational resource, i.e., use only a subset of the available measurements to update (perhaps

a subset of) the states during each Kalman filter cycle, the computation cost could be reduced

dramatically while still maintaining observability over time and improving accuracy per the

SCAAT approach [14]. Specifically we propose using a subset of mσ measurements, where

1 6 mσ ≪ m. The question is which subset of mσ measurements to be used in each cycle.

Previous work suggests pre-determined measurement subsets, however the dynamic nature of

today’s power systems requires more flexibility. By performing a principal component analysis

(PCA) on the error covariance P , we can determine the state subspace to be updated most

urgently (e.g. with larger estimation uncertainty than others).

Because covariance matrices are always symmetric and positive semidefinite, they have

several important properties. Before the measurement update (“Correction" step) begins, let us

consider the PCA of the a priori error covariance matrix P− = U ·D · UT :

1. There exists an orthonormal basis U (UUT = UTU = I , where I is the identity matrix),

whose columns are eigenvectors of P−, such that the error covariance matrix expressed

in this basis is diagonal. The axes of this new basis are called Principal Components of

P−.

2. As the off-diagonal elements of this new diagonal covariance matrix D are zeros, the

new variables defined by this new basis (the projections of the a priori estimate error

e− = x− x− on the Principal Components) are uncorrelated.

3. The diagonal elements of this new matrix D are the eigenvalues of P−. So the variances

of the projections of error e− on the Principal Components are equal to the corresponding

eigenvalues of P−.

4. The eigenvalues in D are decreasingly ordered. The mth eigenvalue corresponds to the

mth eigenvector.

The principal components that correspond to the largest elements of D, indicate the axes

67

in the state space that have the largest estimation uncertainties. Our strategy is to target these

uncertainties first, thus we need to find the set of measurements that can reduce these uncer-

tainties most efficiently. For example, in a 3D tracking application example, if we use several

cameras to estimate the location of a certain object, and we noticed the uncertainty is growing

rapidly in one direction, then in the next cycle we would ideally use a camera which is looking

in an orthogonal direction.

3.6.2 Measurement Selection Procedure

Similar to SCAAT, LoDiM constrains the unknowns over time and refines the estima-

tion continually. Nonetheless, it is the measurement selection that differentiates LoDiM from

SCAAT. In LoDiM, we select the measurements that help our estimate the most (i.e. reduce

estimation uncertainty most effectively) during each iteration.

After the time update (“Prediction" step) of each Kalman filter iteration, we have the n×n

a priori error covariance matrix

P− = U ·D · UT (3.30)

where D is the diagonal matrix consisting of the eigenvalues of P− in decreasing order, and

U is the orthonormal basis whose columns are the corresponding eigenvectors. Note that the

full PCA can also be a time-consuming process, especially if the state space is large. For this

reason, for now we only investigate the first eigenvector (dominant eigenvector) u1 in U , which

represents the directions that we are most uncertain about in the state space.

The dominant eigenpair (the largest eigenvalue and the corresponding eigenvector u1) can

be conveniently computed by existing algorithms such as the well-known power method [40].

Start with a unit vector q(0) ∈ Rn, which is an approximation to the dominant eigenvector, the

68

power method iteratively produces a vector sequence q(k) as follows:

p(k) = P−q(k−1) (3.31)

q(k) = p(k)/∥p(k)∥2 (3.32)

uk = [q(k)]HP−q(k), k = 1, 2, ... (3.33)

Except for special starting points, the iterations converge to the dominant eigenvector u1 quite

efficiently. Table 3.2 has listed the upper bound on computational requirements of the power

method at every iteration, measured by FLOP counts.

Table 3.2: Power Method Computational Cost Upper Bound

Step Variable Computation Breakdown FLOP Breakdown FLOP counts

1 p(k) P−q(k−1) 2n2 − n 2n2 − n

2 q(k) [p(k)]Hp(k) 2n− 1 3n√[p(k)]Hp(k) 1

p(k)/√

[p(k)]Hp(k) n

3 uk [q(k)]HP− 2n2 − n 2n2 + n− 1

[q(k)]HP−q(k) 2n− 1

Adding up the FLOP counts at each step in Table 3.2 yields an estimate of the computational

cost for a power method iteration, which is at most:

(2n2 − n) + (3n) + (2n2 + n− 1) = 4n2 + 3n− 1 (3.34)

Because the power method is an iterative method, one has to stop at some point. Typically one

carries on the process until a convergence criterion of the eigenvalue is satisfied or, alterna-

tively, until a given maximum number of iterations is reached. In our experiments, we follow

the latter approach for each measurement selection procedure (e.g., kmax = 10), such that we

can save the computational cost and include the procedure with each Kalman cycle.

69

Now consider the measurements. We can rewrite the measurement equation as

z = Hx+ v

= HUUTx+ v

= (HU)x′ + v

= H ′x′ + v (3.35)

Where x′ = UTx is the new state vector defined by the new basis U , H ′ is the corresponding

m × n new measurement Jacobian matrix, and v is the unchanged measurement noise vector

with p(v) ∼ N(0, R). Here we assume R to be a m ×m diagonal covariance matrix, i.e., the

measurement noise is uncorrelated.

Because the basis U is composed of unit vectors, H ′ij can also be considered as the magni-

tude of the projection (i.e. the scalar projection) of the ith measurement direction vector in the

direction of the jth basis in U . We now have

Hu1 = [H ′11 H ′

21 H ′31 . . . H

′m1]

T (3.36)

Intuitively, for the same basis, say u1, the larger H ′i1 is, the better the corresponding ith mea-

surement could reduce the uncertainty in this basis direction. However we should also keep

in mind that different measurements have different amounts of noise. Thus for u1 we create a

“ranking" vector r1 from H ′ and R using the following adjustment:

r1 = [H ′2

11

R11

H ′221

R22

H ′231

R33

. . .H ′2

m1

Rmm

]T (3.37)

where R is the measurement noise covariance matrix. The m × 1 vector r1 evaluates the

uncertainty calibration abilities of each measurement regarding the most significant uncertainty

component, scaled by the corresponding measurement noise level. Now we are going to prove

the following:

Lemma 1 The larger H′2i1

Riiis, the more effectively the ith measurement can reduce the uncer-

tainty along direction u1.

70

Proof. In [36], it has been shown that the error covariance update in the measurement update

(“Correction" step)

P = (I −KH)P−

= (I − P−HT (HP−HT +R)−1H)P−

= P− − P−HT (HP−HT +R)−1HP− (3.38)

is equivalent to

P−1 = (P−)−1 +HTR−1H (3.39)

The inverse of the error covariance, P−1, is often called the information matrix. According to

(3.30), we have

P−1 = (UDUT )−1 +HTR−1H

= U−TD−1U−1 + U−TUTHTR−1HUU−1

= U−T [D−1 + (HU)TR−1(HU)]U−1

= U−T (D−1 +H ′TR−1H ′)U−1 (3.40)

Let us denote the matrix (D−1 + H ′TR−1H ′) in equation (3.40) by Σ, then Σ11 is the “infor-

mation" of the new state variable in direction u1, which we were most uncertain about. This

information value is increased/improved from D−111 to Σ11 by

(Hu1)TR−1(Hu1) =

m∑i=1

H ′2i1

Rii

(3.41)

So if we are only willing to incorporate mσ measurements (instead of the full m measurements)

into the measurement update equation, the ones with the largest H′2i1

Riivalues would be our choice.

We can easily locate the mσ measurements with the largest values among these m elements

in vector r1 (3.37). Thus when our budget of computation time and memory space is tight,

we could use only mσ measurements in the next step but still achieve stable estimation results.

71

Using this approach to reduce state estimation uncertainty is similar to fighting the Hydra in

Greek mythology: if we are not able to destroy all the “heads" at once, at least we can aim at

and cut off the most threatening “head" during each round.

3.6.3 LoDiM Architecture

For computational efficiency in large-scale Kalman-filter-based state estimation, we pro-

pose a better realization for our new method, LoDiM, with a parallel task structure.

To generalize our approach, let us consider a non-linear system described by equations

(2.5) and (2.6), with a large measurement-space. LoDiM has its main state estimation cycle,

which is similar to SCAAT algorithm, running on the foreground:

1. Compute the time ∆t since the previous estimate.

2. Predict the state and error covariance. Share the predicted error covariance P− with the

background process. x− = a∆t(xt−∆t , 0)

P− = A∆tPt−∆tAT∆t

+Q∆t

(3.42)

3. If this is the first cycle, choose mσ measurements (which is a much smaller measure-

ment subset) randomly; otherwise, choose the mσ measurements nominated by the back-

ground process. Predict the measurement and compute the corresponding Jacobian. z = hσ(x−t , 0)

H = Hσ(x−t , 0)

(3.43)

4. Compute the Kalman gain.

K = P−HT (HP−k HT +Rσ,t)

−1 (3.44)

72

5. Correct the predicted state estimate and error covariance from (3.42) using the actual

sensor measurement zσ,t. xt = x− +K(zσ,t − z)

Pt = (I −KH)P−(3.45)

Meanwhile, LoDiM has its auxiliary measurement selection process as described in sub-

section 3.6.2, running on the background:

1. Compute the principal component u1 of the error covariance P− predicted on the fore-

ground.

2. Compute Hu1 according to (3.36) and the ranking vector r1 according to (3.37).

r1 = (Hu1). ∗ (Hu1)./diag(R) (3.46)

where .∗ and ./ denote the element-by-element operations.

3. Select the mσ measurements corresponding to the mσ largest elements in r1, to be used

by the foreground process.

Overall, the LoDiM algorithm process can be expressed by the flow chart in Figure 3.3.

73

KF/EKF (foreground) Measurement Selection (background)

!!

Cycle 1

Prediction

Correction (Pick measurements

randomly)

Cycle 2

Prediction

Correction (Pick the selected

measurements)

Cycle 3

Prediction

Correction (Pick the selected

measurements)

Select measurements as

described in Section III-C

Same as above

Same as above

Figure 3.3: The LoDiM algorithm flow chart.

In the future, this algorithm is subject to further improvement by exploiting more task

parallelism and data parallelism using modern parallel computation techniques and hardware.

Another future research topic is that for large systems with exploited sparsity, it makes

sense to integrate LoDiM into hierarchical estimation methods [41–44]. Take large-scale wide-

area power system state estimation as an example, a rational approach could be a hybrid method

which includes the following three major steps:

1. Local State Estimation. A Large-scale wide-area power system can be divided into

smaller non-overlapping subsystems. Local estimation can be done simultaneously and

independently for all subsystems using the LoDiM algorithm.

2. Coordination Vector Estimation. Each subsystem is connected to its neighbors through

tie-lines, which end at boundary buses belonging to these subsystems. This step requires

74

input of tie-line measurements from the synchronized PMUs and part of the voltage mea-

surements from each subsystem (only those of the boundary buses), while the output of

estimated coordination phase angle vector and the local state estimation from all subsys-

tems are accessible for the next step.

3. Global Coordination. All local estimates are coordinated into a unified system-wide

state estimation.


LoDiM Performance on a Smaller System

In this section, we investigate the proposed LoDiM state estimation performance using the

3-machine 9-bus system as shown in Fig. 2.2 from Section 2.8.3. We simulate an emergency

event: a three-phase fault at bus 4 that happens at t = 1, and then cleared in 0.15 seconds. This

event represents a large disturbance emergency.

For this nine-bus system, we simulate the 17 state variables, which are the bus voltage

magnitudes and phase angles, recorded as the “true" system states. There are 54 simulated

Phasor Measurement Unit (PMU) measurements, including both voltage phasor and current

phasor measurements. They are updated every 1/30 second, and combined with 5% random

noise. The algorithm is simulated to run on a processor with 10 megaFLOPS (107 FLoating-

point Operation Per Second) of computing power available.

Without loss of generality, we visualize the state estimation results for bus 2. Fig. 3.4

depicts the voltage magnitude tracking result of bus 2 during the 10 seconds from t = 0 to

t = 10, using the conventional Kalman filter. The black solid line plots the true state, while the

red dot-dot line plots the estimated state. The regular EKF dynamic state estimation uses the

entire measurement set as input at each update step, resulting in lower estimation rate.

75

0 2 4 6 8 100.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

time in seconds

volta

ge m

agni

tude

at b

us 2

in p

er u

nit

estimated |V| using all measurementstrue |V|

Figure 3.4: Bus 2 Voltage magnitude estimation using conventional Kalman filter

Next, we implement a reduced measurement-space state estimation with a naive approach:

a small subset of the measurements (6 measurements in our experiment) are chosen randomly

during each shorter estimation cycle. Fig. 3.5 shows the tracking results using a green dash-dot

line.

76

0 2 4 6 8 100.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

time in seconds

volta

ge m

agni

tude

at b

us 2

in p

er u

nit

estimated |V| using randomly selected measurementstrue |V|

Figure 3.5: Bus 2 Voltage magnitude estimation using randomly chosen measurements

Our LoDiM state estimation method use the same number of dynamically selected mea-

surements (6 measurements also) as input during each short estimation cycle. However, the

small measurement subset is selected by the measurement selection procedure described in

section 3.6.2. Fig. 3.6 shows the voltage magnitude tracking result of the same bus during the

same period, using LoDiM method. The black solid line still represents the true state, while

the blue dash-dash line plots the estimated state.

77

0 2 4 6 8 100.7

0.75

0.8

0.85

0.9

0.95

1

1.05

time in seconds

volta

ge m

agni

tude

at b

us 2

in p

er u

nit

estimated |V| using LoDiMtrue |V|

Figure 3.6: Bus 2 Voltage magnitude estimation using LoDiM

Finally, let us take a closer look at the performances of different state estimation methods

described above. Within a short time period from t = 4 to t = 6, the true state and the

estimated states using these three approaches are plotted in Fig. 3.7 for a better comparison.

This figure illustrates how our proposed LoDiM method, appearing both smooth and accurate,

outperforms the other two methods in this simulation.

78

4 4.5 5 5.5 60.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

time in seconds

volta

ge m

agni

tude

at b

us 2

in p

er u

nit

estimated |V| using all measurementsestimated |V| using randomly selected measurementsestimated |V| using LoDiMtrue |V|

Figure 3.7: The performance comparison of state estimation with full measurement-space (inred), with randomly selected measurements (in green) and with LoDiM (in blue), from t = 4to t = 6

LoDiM Performance on a Larger System

We have also simulated the proposed LoDiM state estimation on the 16-generator 68-bus

system as shown in Fig. 2.6 from Section 2.8.3, which represents the New England/New York

interconnected system. A three-phase fault occurring at time t = 1.1 is simulated on bus 29,

and then cleared in 0.05 seconds.

There are 135 state variables to be estimated in this 68-bus system. We have simulated

468 PMU measurements combined with 5% random noise, including both voltage phasor and

current phasor measurements, at a sampling rate of 30 samples/second. The algorithm is sim-

ulated to run on a processor with 1 gigaFLOPS (109 FLoating-point Operation Per Second)

available computing power.

First, we visualize the state estimate results for bus 60, a generator bus. Fig. 3.8, Fig. 3.9

79

and Fig. 3.10 have illustrated the voltage magnitude tracking results of bus 60 during the 10

seconds from t = 0 to t = 10, using the following different methods:

• a conventional Kalman filter (which takes the entire measurement set as input, resulting

in lower estimation rate);

• a Kalman filter with randomly chosen measurements (which takes a smaller random set

of 70 measurements as input, during each shorter estimation cycle);

• the LoDiM state estimation method (which also takes 70 measurements as input during

each short estimation cycle; nevertheless, the small measurement subset is dynamically

selected by the measurement selection procedure described in section 3.6.2).

0 2 4 6 8 100.86

0.87

0.88

0.89

0.9

0.91

0.92

0.93

0.94

0.95

time in seconds

volta

ge m

agni

tude

at b

us 6

0 in

per

uni

t

estimated |V| using all measurementstrue |V|

Figure 3.8: Bus 60 Voltage magnitude estimation using conventional Kalman filter

80

0 2 4 6 8 100.86

0.87

0.88

0.89

0.9

0.91

0.92

0.93

0.94

0.95

time in seconds

volta

ge m

agni

tude

at b

us 6

0 in

per

uni

t

estimated |V| using randomly selected measurementstrue |V|

Figure 3.9: Bus 60 Voltage magnitude estimation using randomly chosen measurements

0 2 4 6 8 100.86

0.87

0.88

0.89

0.9

0.91

0.92

0.93

0.94

0.95

time in seconds

volta

ge m

agni

tude

at b

us 6

0 in

per

uni

t

estimated |V| using LoDiMtrue |V|

Figure 3.10: Bus 60 Voltage magnitude estimation using our LoDiM method for measurementselection.

Next, we examine a close-up shot for a better comparison. At bus 60, during the short time

81

window from t = 4 to t = 6 seconds, the true state and the estimated states using these three

approaches are plotted in Fig. 3.11. Another similar comparison is performed at bus 26, a load

bus. The zoom-in plot shown in Fig. 3.12 illustrates how the proposed LoDiM algorithm has

improved the state tracking quality.

4 4.5 5 5.5 60.92

0.925

0.93

0.935

0.94

0.945

0.95

0.955

0.96

time in seconds

volta

ge m

agni

tude

at b

us 6

0 in

per

uni

t


Figure 3.11: The performance comparison of different methods at generator bus 60 in the16-generator 68-bus system, with full measurement-space (in red), with randomly selectedmeasurements (in green) and with LoDiM (in blue), from t = 4 to t = 6

82

4 4.5 5 5.5 60.99

1

1.01

1.02

1.03

1.04

1.05

time in seconds

volta

ge m

agni

tude

at b

us 2

6 in

per

uni

t


Figure 3.12: The performance comparison of different methods at generator bus 26 in the16-generator 68-bus system, with full measurement-space (in red), with randomly selectedmeasurements (in green) and with LoDiM (in blue), from t = 4 to t = 6

Optimal Number of Selected Measurements

The number of measurements to be selected in each cycle, mσ, affects the estimation re-

sults. Fig. 3.13 presents the zoom-in bus 2 voltage magnitude tracking results for the smaller

3-generator 9-bus system, from t = 2 to t = 3. Similar estimation behavior is observed in the

larger 16-generator 68-bus system (Fig. 3.14). The true state is plotted using a black solid line.

Our LoDiM state estimation method uses dynamically selected measurements during each cy-

cle as described in section 3.6.2. The blue dash-dash line plots the state when mσ = 6. If mσ is

too small, e.g. mσ = 2, the estimated state resembles a step function as shown in green dash-

dot line. This is because very little information is integrated in each cycle, while the speed-up

of the cycle period is not significant enough to capture the state variation. On the other hand,

the red dotted line plots the estimated state using a conventional Kalman filter, where the entire

measurement set is used as input, resulting in a lower estimation rate.

83

2 2.2 2.4 2.6 2.8 30.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

time in seconds

volta

ge m

agni

tude

at b

us 2

in p

er u

nit

estimated |V|, all measurements per cycleestimated |V|, 2 measurements per cycleestimated |V|, 6 measurements per cycletrue |V|

Figure 3.13: Performance comparison of LoDiM with different mσ values, at bus 2 in the3-generator 9-bus system

2 2.2 2.4 2.6 2.8 30.99

0.995

1

1.005

1.01

1.015

1.02

1.025

1.03

1.035

1.04

time in seconds

volta

ge m

agni

tude

at b

us 2

6 in

per

uni

t

estimated |V|, all measurements per cycleestimated |V|, 20 measurements per cycleestimated |V|, 70 measurements per cycletrue |V|

Figure 3.14: Performance comparison of LoDiM with different mσ values, at bus 26 in the16-generator 68-bus system

84

As depicted above, the LoDiM performance comparison with different mσ values reflects

an interesting trade-off between gained information and processing time. This observation

leads us to believe that there exists a “sweet spot" to achieve optimal performance. To confirm

our supposition, we simulated LoDiM with all possible values of mσ, i.e. from mσ = 1 to full

measurement-space on both power system models.

The average state estimation absolute error (per unit, of all buses, within 10 seconds) of

LoDiM, using different amount of measurements (in percentage), are plotted in blue dash-dash

lines for these two system models respectively in Fig. 3.15 and 3.16. Similarly, the computa-

tional requirements (FLOP counts) for each filter cycle, using different quantities of measure-

ments, are plotted in green dot-dot lines. Optimally adapting to the available computing power,

the percentage of measurements to be selected during each cycle is 76% for the small system,

and 27% for the large system.

0 20 40 60 80 1000.025

0.03

0.035

0.04

0.045

0.05

Est

imat

ion

Abs

olut

e E

rror

in p

er u

nit

Measurement selected per cycle in percentage

0 20 40 60 80 1000

1

2

3

4

5

6

7x 10

5

FLO

P c

ount

s

Average Absolute ErrorFLOPs per KF cycle

Figure 3.15: LoDiM performance versus different amount of selected measurements in the3-generator 9-bus system, with 10 megaFLOPS computing power available

85

0 20 40 60 80 1000.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

Est

imat

ion

Abs

olut

e E

rror

in p

er u

nit

Measurement selected per cycle in percentage

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

8

FLO

P c

ount

s

Average Absolute ErrorFLOPs per KF cycle

Figure 3.16: LoDiM performance versus different amount of selected measurements in the16-generator 68-bus system. with 1 gigaFLOPS computing power available

3.7 My Method: Reduced Measurement-space Dynamic StateEstimation (ReMeDySE) in Power Systems

Modern phasor measurement technologies provide high-speed sensor data with precise time

synchronization for power system analysis, motivating us to consider including dynamic states

in the state estimation process, e.g., generator rotor angles and generator speeds, along with the

static state elements of bus voltage magnitudes and phase angles.

Applying Kalman filter techniques to dynamic state estimation is a developing research area

in modern power systems. For example, [11] investigated the feasibility of applying Kalman

filter techniques to include dynamic state elements in the state estimation process. Compared to

traditional steady state estimators, the Kalman filter is able to track dynamic state variables ef-

ficiently and accurately. However, in large-scale and wide-area interconnected power systems,

the combination of high computational complexity (primarily due to the very large number of

measurements) and slow processing speed (primarily due to limited computational resources)

86

presents a significant challenge. To help address this challenge, we have derived the Reduced

Measurement-space Dynamic State Estimation (ReMeDySE) from the LoDiM algorithm in

Section 3.6, for power system dynamic state estimation. Together with the fast advancing pha-

sor measurement technologies, the Kalman-filter-based power system dynamic state estimation

should be plausible to implement in on-line systems.

3.7.1 The Power System Dynamic State Space

The specific state elements we consider here are generator rotor angles and generator

speeds. Without loss of generality, in a power system that consists of n generators and m

buses, let us consider the generator i which is connected to the generator terminal bus i. We

use a classical model for the generator composed of a voltage source |Ei|∠δi with constant am-

plitude behind an impedance X ′di

. The nonlinear differential-algebraic equations for generator

i can be written asdδidt

= ωB(ωi − ω0)

dωi

dt= ω0

2Hi(Pmi

− |Ei||Vi|X′

di

sin(δi − θi)−Di(ωi − ω0))(3.47)

where state variables δ and ω are the generator rotor angle and speed respectively, ωB and ω0

are the speed base and the synchronous speed in per unit, Pmiis the mechanical input, Hi is

the machine inertia, Di is the generator damping coefficient and |Vi|∠θi is the phasor voltage

at the generator terminal bus i.


change in state can be modeled by the differential equation

dx

dt= Acx+ wc, (3.48)

where wc is an 2n × 1 continuous time process noise vector with 2n × 2n noise covariance

matrix Qc = E[wcwTc ], and Ac is a 2n× 2n continuous time state transition Jacobian matrix.

87

Section 2.9 describes the detailed derivations of the corresponding discrete-time state tran-

sition matrix A and discrete-time process noise covariance Q (also see [11] and [8] for addi-

tional information). According to the process model from equation (3.1) we have

xk = Axk−1 + wk−1 (3.49)

= (I + Ac ·∆t)xk−1 + wk−1, (3.50)


Power system measurements include bus voltage magnitudes, bus voltage angles and line

flows. The measurement equations can be derived using the nodal admittance matrix and bus

voltage reconstruction matrix, according to the expanded system nodal equation:

Yexp

E

V

=

YGG YGL

YLG YLL

E

V

=

IG

0

(3.51)

where E = [|E1|∠δ1, |E2|∠δ2, ..., |En|∠δn]T is the vector of internal generator complex volt-

ages, V = [|V1|∠θ1, |V2|∠θ2, ..., |Vn|∠θn]T is the vector of bus complex voltages, IG represents

electrical currents injected by generators, Yexp is called the expanded nodal matrix, which in-

cludes loads and generator internal impedances: YGG, YGL, YLG and YLL are the corresponding

partitions of the expanded admittance matrix. Therefore the relationship of bus voltages V to

generator voltages E can be expressed as:

V = −Y −1LL YLGE = RVE (3.52)

where RV is defined as the bus reconstruction matrix. In a power flow model, the line flows

are functions of bus voltages V , and hence can be easily constructed as functions of generator

voltages E using (3.52).

The measurement model, as discussed in Section 2.9 (also see [11] and [8] for additional

information), can be expressed in the form of equation (3.2) after linearization

z = Hx+ v (3.53)

88

where H is the corresponding Jacobian matrix

H =

∂h1

∂δ10 ∂h1

∂δ20 · · ·

∂h2

∂δ10 ∂h2

∂δ20 · · ·

∂h3

∂δ10 ∂h3

∂δ20 · · ·

∂h4

∂δ10 ∂h4

∂δ20 · · ·

......

...... . . .

, (3.54)

and v is the measurement noise with normal probability distribution p(v) ∼ N(0, R). The

measurement noise covariance R can be determined experimentally, by off-line testing of the

measurement devices over time and (if desired) under different conditions.

Note that the measurement Jacobian H has all zero entries in even-numbered columns.

Thus in the measurement selection procedure (previously described in Section 3.6.2), we can

save the computational cost by performing PCA on an a priori error covariance submatrix P−s ,

consisting of elements of the odd-numbered rows and odd-numbered columns of P−. The com-

putational cost of this procedure is further reduced, because the corresponding measurement

Jacobian submatrix Hs is then formed by only odd-numbered columns of H .


We used the ReMeDySE approach on the standard 16-generator 68-bus model (Fig. 2.6)

representing the New England/New York interconnected system. We simulated a three-phase

fault at bus 29, starting from t = 1.1 and lasting for 0.05 seconds, as a dynamic event repre-

senting a large disturbance emergency.

We simulated the dynamic state variables (the rotor angles and speeds of each generator)

and recorded them as the actual dynamic states. Here we made the strong assumption that

all bus voltages and phase angles are measured by PMUs. In the future we will relax this

assumption. These simulated PMU measurements contain 1% random noise this time.

89

Fig. 3.17 depicts the actual and the estimated speed (using ReMeDySE with only 15%

measurements per cycle, and regular EKF with all PMU measurements per cycle) of generator

2 within 10 seconds from t = 0 to t = 10, to capture its reaction to this large disturbance.

0 2 4 6 8 100.997

0.998

0.999

1

1.001

1.002

1.003

t in seconds

gene

rato

r sp

eed

in p

u

estimated speed with ReMeDySEestimated speed with regular EKFactual speed

Figure 3.17: Estimation of rotor speed at generator 2 using ReMeDySE and regular EKF, witha 3-phase fault at bus 29

For a better comparison, Fig. 3.18 zooms in the green dashed-line window, and gives us a

close-up look of the short window from t = 2 to t = 5 of our simulation. Similarly, Fig. 3.19

shows the rotor angle estimation results at this same generator during the same window.

90

2 2.5 3 3.5 4 4.5 50.998

0.9985

0.999

0.9995

1

1.0005

1.001

1.0015

1.002

t in seconds

gene

rato

r sp

eed

in p

u


Figure 3.18: Performance comparison ofReMeDySE and regular EKF estimating rotorspeed at generator 2, from t = 2 to t = 5

2 2.5 3 3.5 4 4.5 520

25

30

35

40

45

50

55

t in seconds

gene

rato

r ang

le in

deg

rees

estimated angle with ReMeDySEestimated angle with regular EKFactual angle

Figure 3.19: Performance comparison ofReMeDySE and regular EKF estimating rotorangle at generator 2, from t = 2 to t = 5

Take generator 10 as another example, Fig. 3.20 and Fig. 3.21 illustrate the rotor speed and

angle estimation results at generator 10 respectively, during the same period of time.

2 2.5 3 3.5 4 4.5 50.998

0.9985

0.999

0.9995

1

1.0005

1.001

1.0015

1.002

t in seconds

gene

rato

r sp

eed

in p

u


Figure 3.20: Performance comparison ofReMeDySE and regular EKF estimating rotorspeed at generator 10, from t = 2 to t = 5

2 2.5 3 3.5 4 4.5 520

25

30

35

40

45

t in seconds

gene

rato

r ang

le in

deg

rees

estimated angle with ReMeDySEestimated angle with regular EKFactual angle

Figure 3.21: Performance comparison ofReMeDySE and regular EKF estimating rotorangle at generator 10, from t = 2 to t = 5

These plots demonstrate that under the experimental conditions, our proposed ReMeDySE

approach outperforms the traditional EKF approach with higher accuracy. The primary reason

is that the standard batch approach for EKF-based dynamic state estimation uses the entire

91

measurement set as input at each step, resulting in lower estimation rate and higher latency.

The dynamic nature of the state variables introduces additional noise (uncertainty) into the

process, and this noise increases as the latency increases. Typically when latency is small, the

measurement device noise dominates; when latency is large, the process noise dominates. In

ReMeDySE, only a small and dynamically selected subset of PMU measurements is used, to

adapt to the limited computational resource. Our method, combined with developing PMU

technologies, makes high-speed high-accuracy and high-flexibility dynamic state estimation

realizable in future power systems.

92

CHAPTER 4

ADAPTIVE AND ROBUST ESTIMATION

Kalman filters have been successfully applied in many different research areas over the last

five decades. In modern power systems, especially with the emergence of real-time digital me-

ters, they are commonly seen in system state tracking methods to enable real-time monitoring

and control.

Kalman filters achieve optimal performance only when the system noise characteristics

have known statistical properties (zero-mean, Gaussian, and spectrally white). However in

practice the process and measurement noise models are usually difficult to obtain. Thus we

propose an efficient adaptive Kalman filter algorithm, that can identify and reduce the impact

of incorrect system modeling and/or bad PMU measurements. We call this novel approach

Adaptive Kalman Filter with Inflatable Noise Variances (AKF with InNoVa). Simulations

demonstrate its robustness to sudden changes of system dynamics and erroneous measure-

ments.

Utilizing AKF with InNoVa as a fundamental building block, later we develop a real-time

two-stage Kalman filter approach [45]: at every time step, stage one takes the raw PMU data

into the AKF with InNoVa, to estimate the traditional states (bus voltage phasors); then stage

two feeds the results into an Extended Kalman filter (EKF) to estimate the truly dynamic states

(generator rotor angles and speed). Our method is efficient, effective and robust under various

simulated scenarios.

93

4.1 Introduction

As electricity demand continues to grow and renewable energy increases its penetration

in the power grid, real-time state tracking becomes essential for system monitoring and con-

trol. Nowadays, the high-speed time-synchronized data provided by PMUs is making real-time

power system state estimation possible.

The Kalman filter is emerging as a popular mechanism for estimating such power system

states dynamically, in conjunction with real-time phasor measurement technologies. Ideally,

the employed process and measurement models would represent the “true" systems, with ac-

curate statistical noise characteristics. In practice one has to develop models for the systems,

whereas all such models are more or less sub-optimal.

Fortunately, as statistician George Box once said, “All models are wrong, but some are

useful" [46]. With these “useful" models, Kalman filter are able to estimate states even when

the precise nature of the modeled system is unknown. Nevertheless, when the system experi-

ences unpredictable process changes, and/or the measurements contain significant errors, the

estimated states could deviate from the true states rapidly.

A lot of efforts have been made to improve Kalman filters under different circumstances

[47–50]. Here we propose a more general approach: Adaptive Kalman filter with Inflatable

Noise Variances (AKF with InNoVa), to achieve more robust state estimation. By simultane-

ously adjusting noise parameters Q and R on-the-fly, this novel AKF approach is remarkably

efficient in dealing with inaccurate system models and un-modeled measurement errors. Tak-

ing one step further, we propose a real-time two-stage Kalman filter approach, for estimating

the traditional states as well as the dynamic states in a power system. At every time step, stage

one takes the raw PMU measurements into the AKF with InNoVa, to estimate the traditional

states; then stage two takes the results into an Extended Kalman filter (EKF) to estimate the

truly dynamic states. Both stages produce state estimates that are critical in various power

94

system applications.


In the rest of this chapter, Section 4.3 introduces the background. Section 4.4 briefly de-

scribes the traditional methods for bad data detection and identification. Section 4.5 presents

the principles and implementation details of our AKF with InNoVa. Built on this adaptive

Kalman filter algorithm, the two-stage Kalman filter approach is proposed in Section 4.6.


Traditionally, a weighted least square (WLS) estimator is the classic method for power

system state estimation. In WLS state estimation, methods such as Largest Normalized Resid-

ual Test and Hypothesis Testing Identification are proposed and extensively used for bad data

detection and identification [21]. For example, in [51] the authors study the usage of phasor

measurements in WLS state estimation, and how to identify bad data with normalized residual

vectors. In [50], although the authors proposed a robust extended Kalman filter (REKF) for

dynamic harmonic state estimation, which can detect the presence of sudden load change or

suspicious measurements with normalized EKF innovation vectors, they still use normalized

WLS residual vectors to confirm bad measurements.

WLS estimators are considered “static" because at any one time instant, the state is cal-

culated only from the measurement set at that time. On the other hand, if an estimator can

predict the state ahead of time, it is considered “dynamic". [33] gives an overview on different

methodologies and developments in power system tracking and dynamic state estimation based

on previous research.

95

Although the term dynamic state estimation (DSE) can be traced back to the 1970s [32],

it was the development of phasor measurement technologies since the 1980s [2] that made it

possible to capture the dynamic behavior of the system. Most DSE approaches are Kalman-

filter-based, despite the fact that a reliable system state evolution (process) model is usually not

available. To overcome this situation, the approach proposed in [52] predicts the load flow data

at the next time instant using a short-term nodal load forecasting technique, and combines the

PMU data at the next time instant for DSE. Another approach is presented in [53], where the

authors use historical system state data and the Holt’s double exponential smoothing method to

estimate process model parameters such as the state transition matrix and the trend component.

[54] presents more detailed descriptions of these approaches and a comprehensive survey of

other state estimator formulations in the presence of PMUs.

A major challenge with DSE using PMUs is that large-scale measurement data needs to

be processed in a very short period of time (short system update cycle). This is one reason

why dynamic adaptation of noise modeling parameters seems attractive. Previous studies have

demonstrated satisfying tracking results by adjusting noise parameters, under different condi-

tions. For example, to deal with unpredictable process changes, [47] has proposed a reverse

prediction adaptive Kalman filter algorithm, which adjusts the process noise covariance Q to

improve filter precision, assuming the measurement model is correct. As a comparison, [48]

adjusts the measurement noise covariance R instead of Q, to increase the robustness against er-

roneous measurements. Assuming the a priori noise statistics Q and R are both unknown, the

authors of [49] constructed the autocorrelation functions of innovation sequence, and incorpo-

rated information about the quality of the autocorrelation parameters through a WLS method.

The latter permits to generate a convergent sequence to the steady state filter, which allows to

determine Q and R after some manipulations.

In our approach, we have in mind the existence of unexpected process changes as well

as measurement errors. Our proposed method, Adaptive Kalman Filter with Inflatable Noise

96

Variances (AKF with InNoVa), is able to adjust noise statistics Q and R on-the-fly by analyzing

the innovation and residual vectors. These inflatable (and deflatable) noise variances are the

keys to our algorithm’s robustness.

Furthermore, as mentioned above, [11] and [34] utilized an extended Kalman filter (EKF)

based approach to estimate the dynamic states of power systems. This approach can achieve

satisfying and robust performance in terms of tracking the dynamic system states, with the

assumption that all PMU measurements are ideal, and the hypothetical models are the “true"

models, with accurate noise statistical characteristics. Now with AKF with InNoVa, we can

relax this assumption: our two-stage Kalman filter approach is able to take in inaccurate system

models and erroneous measurements, and produce robust estimation of the traditional states

as well as the dynamic states in a power system, given sufficient data redundancy.

4.4 Traditional Bad Data Detection and Identification

In practice, even with accurately modeled systems, unexpected bad measurements can oc-

cur due to various reasons. Thus people have developed many techniques to handle these bad

data. Once detected, they need to be identified and eliminated or corrected, to support unbiased

state estimation.

Considering the state space models described in Section 2.4 and the WLS state estimator

described in Section 2.6, we will briefly introduce some traditional bad data detection and

identification methods in the following subsections.

97

4.4.1 Chi-squares χ2-test

One of the most common methods for detecting bad data is Chi-squares (χ2) test. It is

comprised of the following steps:

1. After solving the WLS estimation problem, compute the objective function J(x) from

equation (2.15):

J(x) =m∑i=1

(zi − hi(x))2

Rii

=m∑i=1

(eNzi )2 (4.1)

where x ∈ Rn is the estimated state vector, h(x) ∈ Rm is the estimated measurement

vector, z ∈ Rm is the measurement vector, (z−h(x)) is the measurement residual vector,

Rm×m is the measurement error covariance matrix.

Note that here we are approximating measurement error (z − h(x)) by measurement

residual (z−h(x)), and at least n measurements need to satisfy power balance equations,

so at most (m − n) of the normalized measurement “errors" eNzi will be independent

random variables with Standard Normal Distribution N (0, 1). Thus J(x) is assumed to

have a χ2 distribution with (m− n) degrees of freedom.

2. Test the value of J(x):

J(x)

≥ χ2(m−n,p), bad data detected

< χ2(m−n,p), no bad data

(4.2)

The threshold χ2(m−n,p) can be looked up from the χ2 distribution table, corresponding to

a pre-specified detection confidence level with probability p.

χ2 test is computationally inexpensive, however the trade-off is its inaccuracy. It may fail to

detect bad data due to the approximation of measurement errors by measurement residuals in

equation (4.1).

98

4.4.2 Largest Normalized Residual Test

To promote accuracy, one of the most popular bad data identification (and inherently bad

data detection) method, the Largest Normalized Residual (rNmax) Test method, was proposed

with the following steps:

1. After solving the WLS estimation problem, compute the normalized residual vector rNz

with each element

rNzi =|zi − hi(x)|√

Ωii

, i = 1, 2, ...,m (4.3)

where Ωm×m is the measurement residual covariance matrix calculated as follows

Ω = R−H(HTR−1H)−1HT (4.4)

The detailed proof of Cov(rz) = Ω can be found in many technical literatures such as

[21].

2. If the largest element in the normalized residual rNz is larger than the pre-specified identi-

fication threshold (e.g., 3.0), the corresponding measurement is suspected as bad data, the

procedure continues to step 3. Else, the procedure concludes with no bad data suspected.

3. Eliminate the suspected measurement from the measurement set and go to step 1.

The (rNmax) test can perform the detection and identification processes simultaneously using

normalized residuals. However, the measurement residual covariance matrix Ω is more com-

putationally expensive, thus χ2 test is often used for preliminary bad measurement detection.

4.5 My Method: Adaptive Kalman Filter with Inflatable Noise Variances(AKF with InNoVa)

This section first briefly discusses the system modeling of the conventional power system

state estimation problem, with PMU measurements. Then we give a detailed description of the

99

Adaptive Kalman filter with Inflatable Noise Variances algorithm.

4.5.1 System modeling

With the future power systems in mind, we assume the measurements are high-speed syn-

chrophasors provided by PMUs [55]. Because the relationship between the measurements and

the states can be effectively linear, the rectangular coordinate formulation will be preferable

[54], where the real and imaginary parts of bus voltages are considered state variables. This

formulation can prevent numerical problems encountered during flat start when using current

phasors. For the process model, the power system is assumed stable, hence a quasi-static power

system model is employed in our simulations (note that this could be a wrong assumption).

Due to resource limitations, engineers typically have to place PMUs selectively at scat-

tered locations to cover the entire system [8, 56, 57], resulting in low data redundancy level.

Nonetheless, the redundancy can be increased if we treat the state predictions in our dynamic

estimation method as another input source. Here we assume that a PMU installed at a spe-

cific bus can measure the bus voltage phasor, as well as the current phasors along all the lines

incident to the bus. The modeling details can be found in our previous research [8] (also see

Section 2.8).

4.5.2 Adaptive Kalman Filter Algorithm

Recall that we have introduced Kalman filters in Section 3.4. The traditional Kalman filter

is actually a recursive solution to the discrete-data linear filtering problem. It estimates a pro-

cess by using a form of feedback control: the filter estimates the process state at some time and

then obtains feedback in the form of noisy measurements.

100

The equations for the Kalman filter fall into two groups—time update equations (“Predic-

tion") and measurement update equations (“Correction"):

Prediction :

x−k = Axk−1

P−k = APk−1A

T +Q(4.5)

Correction :

Kk = P−

k HT (HP−k HT +R)−1


k )

Pk = (I −KkH)P−k

(4.6)

We define the innovation vector, I−k = (zk −Hx−k ), to be the difference between the true

measurement vector and that computed from the a priori state estimate. The corresponding

innovation covariance is Sk = HP−k HT +R [36].

Although some authors use the terms “innovation" and “residual" interchangeably, they

should not be confused here. Throughout we refer to I−k = (zk − Hx−k ) as the innovation,

to distinguish it from the residual Ik = (zk −Hxk), which is the difference between the true

measurement vector and that computed from the a posteriori state estimate [58].

Lemma: Ideally, the residual vector Ik = (zk −Hxk) should be normally distributed with

zero mean and covariance RS−1k R.

Proof. According to the a posteriori state estimate in (4.6) by incorporating the measurement,

we have


k )

Hxk = Hx−k +HKk(zk −Hx−

k )

zk −Hxk = (zk −Hx−k )−HKk(zk −Hx−

k )

zk −Hxk = (I −HKk)(zk −Hx−k ), (4.7)

101

where I is the identity matrix. Then the mean of the residual is

E(zk −Hxk) = (I −HKk)E(zk −Hx−k ) = 0, (4.8)

and the covariance of the residual is

cov(zk −Hxk) = (I −HKk)cov(zk −Hx−k )(I −HKk)

T

= (I −HKk)Sk(I −HKk)T . (4.9)

Next we will show that I −HKk = RS−1:

I −HKk = I −HP−k HT (HP−

k HT +R)−1

= R(HP−k HT +R)−1

= RS−1k . (4.10)

Combining (4.9) and (4.10), and we can write

cov(zk −Hxk) = RS−1k Sk(RS−1

k )T

= R(S−1k )TRT

= RS−1k R, (4.11)

because the innovation covariance Sk and measurement noise covariance R are both symmetric

matrices. We define Tk = RS−1k R as the residual covariance. Based on the lemma proved

above, we propose our new algorithm: Adaptive Kalman Filter with Inflatable Noise Variances

(AKF with InNoVa).

In the AKF we treat all un-modeled measurement errors as noise. By definition, these

errors are unknown and unpredictable, so they cannot be reflected in the measurement noise

covariance R. Similarly, we also treat deviations from the true process model as un-modeled

noise that is not reflected in the process noise covariance Q. For now we do not consider cases

with multiple/switching process models.

102

To assess the performance of a filter, traditionally people examine the innovation I−k =

(zk−Hx−k ), which should have a normal distribution with zero mean and covariance Sk. When

the implemented system model does not match reality, the mean of the innovation can shift,

and the magnitude cam grow, so that eventually some element(s) in the normalized innovation

vector (normalized by its covariance Sk) will exceed a predetermined threshold. Usually, the

corresponding measurement(s) are identified as the outliers, and are excluded from further

computation. However, due to the nature of Kalman filter (Figure 4.1), it is usually impossible

to determine whether the shift/growth is caused by a system model mismatch, a measurement

error, or both. If it is the first case, we certainly do want to preserve those apparent “outlier"

measurements.

Figure 4.1: High-level diagram of Kalman filter, courtesy of Greg F. Welch. System modelerror, and/or measurement error, can all lead to significant state estimation error when passingthrough a Kalman filter

In other words, it is insufficient to determine the “outliers" by solely examining the in-

novations, because Q and R are already blended (indistinguishable) in Sk. This is why we

want to investigate the residual Ik = (zk −Hxk) as well: its ideal covariance Tk can be used

to identify the un-modeled measurements errors. Thus instead of simply discard suspicious

measurements, this pair of innovation-residual tests allow us to give them a second chance.

The basic idea of our approach is as follows. To simplify the notation we omit the time

103

step count k. Within each cycle, after the prediction step, we compute the ideal innovation

covariance S and the normalized innovation vector I− where

I−i = |I−i |/√

Sii, i = 1, 2, ...n. (4.12)

If I−i > τ for some threshold τ , then i ∈ Out, where Out holds the outlier indices. In our

experiments, we used τ = 3 (measurement units).

First we assume the anomaly are caused by unknown process noise, so we want to “inflate"

Q by a diagonal matrix ∆Q, so that P− is also inflated by ∆Q. Thus S is consequently inflated

by ∆S = H(∆Q)HT and

I−i = |I−i |/√

Sii +∆Sii ≤ τ, i = 1, 2, ...n. (4.13)

It is easy to show that

∆Sii =n∑

j=1

H2ij∆Qj = (H(i, :) ·H(i, :))([∆Q1, ...∆Qn]

T ), (4.14)

where “·" denotes dot product. We could simply use linear programming method to solve an

optimization problem:

minn∑

j=1

∆Qj (4.15)

s.t. ∆Sii = (H(i, :) ·H(i, :))([∆Q1, ...∆Qn]T )

≥ (|I−i |/τ)2 − Sii, ∀i ∈ Out

∆Q1 ≥ 0,∆Q2 ≥ 0, ...∆Qn ≥ 0.

The inflated Q is then incorporated in the correction step. Similarly, we compute the ideal

residual covariance T and the normalized residual vector I where

Ii = |Ii|/√Tii, i = 1, 2, ...n. (4.16)

If Ii > τ , then i ∈ MeasOut where MeasOut holds the measurement outlier indices, indi-

cating abnormal measurements.

104

Now we can separate the measurement “noise" from the process “noise". Let us denote

ProcOut = Out\MeasOut = i : i ∈ Out and i /∈ MeasOut as the set difference

between Out and MeasOut. If MeasOut is not empty, we will recalculate the ∆Q as in

optimization problem (4.15), except only for ∀i ∈ ProcOut, and update Q to Q+∆Q. As for

the measurements, we “inflate" R in this way: for ∀i ∈ MeasOut, Rii is inflated to λiRii. As

a result, T = RS−1R is also inflated so that the ith diagonal element is now λ2iTii and

Ii = |Ii|/λi

√Tii ≤ τ, i = 1, 2, ...n. (4.17)

It is very easy to compute λis by

λi = (|Ii|/√

Tii)/τ, i ∈MeasOut. (4.18)

Finally, with the inflated Q and R, we recompute the correction step (4.6) to obtain a more

robust state estimation. The Q and R are also updated and carried on to the next cycle.

Furthermore, Q and R should also be deflatable if the abnormal process/measurement prob-

lems are only temporary and eventually resolved. Thus we adopt an exponential decay in the

process model to enable automatic deflation of the parameters. The decay speed can be cus-

tomized by users, according to their expectation and the specific circumstances. For instance,

the user can set a decay constant r, such that

Qk = Qk−1 ∗ e−r, Rk = Rk−1 ∗ e−r (4.19)

To sum up, the AKF with InNoVa has the following steps within each cycle:

1. Predict the state and error covariance using time update equations (4.5).

2. Compute the normalized innovation vector I− as in (4.12). Determine the set of outliers

Out by collecting any i such that I−i exceeds the threshold τ . If Out = ∅, deflate Q and

R as in (4.19) and go to step 8. Else, continute.

3. Construct the optimization problem (4.15) for ∀i ∈ Out and solve for ∆Qi.

105

4. Compute the normalized residual vector I as in (4.16). Determine the set of measure-

ment outliers MeasOut by collecting any i such that Ii exceeds the threshold τ , and

the set of process outliers ProcOut = Out\MeasOut by taking the set difference. If

MeasOut = ∅, go to step 7. Else, continue.

5. Reconstruct the optimization problem (4.15) for ∀i ∈ ProcOut and solve for ∆Qi.

6. Inflate measurement noise variances Rii to λiRii, where λi is computed in (4.18).

7. Inflate process noise variances Qii to Qii +∆Qi.

8. With updated Q and R, correct the state and error covariance using measurement update

equations (4.6) and go to step 1.

4.5.3 Algorithm Performance Evaluation

In order to compare our proposed algorithm against other Kalman filter algorithms, we have

considered two measures of accuracy that are most commonly used in quantitative methods of

forecasting:

• Mean Absolute Error, denoted by MAE. At each time step, it is calculated in the following

way:

MAE =1

n

n∑i=1

|xi − xi| =1

n

n∑i=1

|ei| (4.20)

where xi is the estimated state at bus i, xi is the actual state at bus i, and ei = |xi − xi|

is the absolute error.

• Mean Absolute Percentage Error, denoted by MAPE. At each time step, it is calculated

as follows:

MAPE =1

n

n∑i=1

| xi − xi

xi

| × 100% =1

n

n∑i=1

| eixi

| × 100% (4.21)

106

where xi is the estimated state at bus i, xi is the actual state at bus i, and ei = |xi − xi|

is the absolute error.

Mean Absolute Percentage Error is the mean of the sum of all percentage errors for a

given data set taken without regard to sign. In other words, the absolute relative errors are

summed and the average computed. Comparing to MAE, MAPE does not depend on the

scaling of the variable, which may be inconvenient if the criteria are used for comparing

predictive accuracy across different variables or different time ranges. However, MAPE

achieves scale independence by a simple division by xi. In practical applications, this

entails two major problems:

– If there are zero values, there will be a singularity problem. The contribution of bus

i at that time point and hence the MAPE are undefined.

– Even if xi is only approximately zero, the relative contribution of bus i will be

enormous. Usually, there is no justification for preferring a high precision for small

values of xi. When there is a perfect fit, MAPE is zero. However MAPE has no

restriction on its upper level.

In our application of power system analysis, we have adopted a per-unit system, in which

the system quantities are expressed as fractions of a defined base unit quantity. Besides, when

a three-phase fault happens, the voltage at that bus can actually drop to a value that is very

close to zero. Taking all aspects into consideration, we have decided to use MAE to evaluate

the state estimation accuracy of each algorithm.

4.5.4 PMU Measurement Evaluation

According to IEEE C37.118.1-2011 — IEEE Standard for Synchrophasor Measurements

for Power Systems [55], the measurement of a PMU must maintain certain phasor accuracy

107

over a range of operating conditions. The theoretical phasor value of a signal being measured

and the estimated values obtained from a PMU may include differences in both magnitude

and phase angle. These differences have been combined into a single error quantity called the

“Total Vector Error" or TVE. The standard has defined this quantity as the measure of accuracy

for PMUs.

The criterion TVE is an aggregated difference between a “perfect" sample of a theoretical

synchrophasor and the estimate given by the PMU under test at the same time instant. It is

defined as the square root of the difference squared between the real and imaginary parts of the

theoretically “true" phasor and the estimated phasor, ratioed to the magnitude of the theoretical

phasor and presented as a percentage as seen in Equation (4.22). In this way, the value is

normalized and expressed as per unit of the theoretical phasor.

TV En =

√(Re(xn)−Re(xn))2 + (Im(xn)− Im(xn))2

(Re(xn))2 + (Im(xn))2∗ 100% (4.22)

where Re(xn) and Im(xn) are the estimated phasor values given by the unit under test at time

instant n, Re(x)n and Im(x)n are the true phasor values of the input signal at time instant n.

Note that as with most measurements, the “true" value can never be precisely known, thus we

rely on calibration to establish the bounds within which the measurement (the vector) has a

high probability of ocurring.

A 1% criterion established by setting TV E = 1% in Equation (4.22) can be visualized as a

small circle drawn on the end of the phasor vector. The maximum magnitude error is 1% when

the error in phase angle is zero, and the maximum error in angle is about 0.573 degrees. Fig.

4.2 shows the circle, whose size is greatly exaggerated for clarity.

108

Figure 4.2: The circular boundary of a 1% TVE criterion


We have simulated the performance of the proposed adaptive Kalman filter algorithm (AKF

with InNoVa) on several multi-machine systems with PMU measurements, under various con-

ditions. For better illustration, we will show the simulated results from the 3-generator-9-bus

system, Fig. 2.2, as seen in Section 2.8.3. In total 5 seconds are simulated in time steps of 0.01

seconds. The bus voltages are stable until an unexpected emergency event occurs: a three-

phase fault at bus 6 happens at t = 1 second, and is then cleared in 0.15 seconds. This event

represents a large disturbance emergency, causing the oscillations in all bus voltages. The sim-

ulated PMU data contains random noise where TV E = 5%. Under ideal conditions, where

system models and noise statistics match the reality and no measurement errors, our AKF with

InNoVa performs exactly the same as the traditional KF. The normal cases are omitted here,

because we are more interested in the filter performance under adverse circumstances.

We have simulated different abnormal test cases to illustrate the robustness of our proposed

algorithm. In each case, we executed three Kalman filter algorithms for a more informative

comparison: one is the traditional Kalman filter; another is a naive robust Kalman filter (RKF)

109

which does not adjust model parameters, but uses the largest normalized innovation test to iden-

tify and exclude bad measurements; the last one is our proposed AKF with InNoVa algorithm.

We visualize the voltage tracking results with three-dimensional, real-part and imaginary-part

plots separately.

Then we illustrate the MAE plots and their histograms for a more comprehensive compari-

son. In this particular application, the states are complex voltages. Thus the MAE can also be

written as:

MAE =1

n

n∑i=1

∥xi − xi∥ =1

n

n∑i=1

∥ei∥ (4.23)

where ∥ ∥ denotes the norm (magnitude) of the complex values. To maintain consistency of

our experiments in this subsection, the data range of each histogram is set to be [0, 1], and the

bin (i.e. interval) size is set to be 0.01. These test cases are stated below.

Case 1: under-valued initial Q and R

In this case, the process noise variance of each state variable in per unit is initialized to be

1e−6, resulting in a quite small process noise covariance Q, which means the modeler is very

confident that the system will be stable. Also, the PMU measurement noise is initialized to be

3% only, while it is actually simulated to be 5% Total Vector Error.

Without loss of generality, we will show the voltage tracking results of the three filters for

bus 4 in Fig. 4.3. In this and the following simulations, we always use a black solid line to plot

the true state, a red dash-dash line to plot the traditional KF estimated state, a green dash-dot

line to plot the naive RKF estimated state, and a blue dot-dot line to plot the state estimated by

our AKF with InNoVa.

From Fig. 4.3 we can tell that the KF and the naive RKF are both strongly affected by the

small Q: the estimated states have slow and small oscillations. On the other hand, it is obvious

110

that the AKF with InNoVa tracks closely to the true state. We have observed that in as little as

5 seconds, the process noise variance at bus 6 has grown to 0.1128, which is the largest among

all noise variances. This is no coincidence because the fault occurs at bus 6 exactly. Also, the

average measurement noise becomes 4.8%, which is much closer to the ideal 5%.

Next, the state estimation accuracy of the three filters is evaluated using the MAE plots and

their histograms, and plotted in Fig. 4.4(a) and Fig. 4.4(b).

Fig. 4.4 illustrated how our AKF with InNoVa has achieved the highest accuracy. Not as

robust as its name might suggest, the naive RKF has even larger MAE values than the traditional

KF after the fault occurs. The reason is that the naive RKF, being totally unaware of its poor

estimating performance, is in fact largely caused by a very small Q initialization, and has

dropped some measurements it falsely suspected. Hence its estimation results have drifted

even further from the true states.

111

12

34

5

−1

−0.5

0

0.5

1−1

−0.5

0

0.5

1

Time in seconds

Bus 4 voltage estimation

Bus 4 voltage in pu, real

Bus

4 v

olta

ge in

pu,

imag

inar

y

True stateTraditional KFNaive RKFAKF with InNoVa

(a) Bus 4 voltage estimated by three filter in case 1

0 1 2 3 4 5

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time in seconds

Bus

4 v

olta

ge in

pu,

rea

l

Bus 4 voltage estimation, real part


0 1 2 3 4 5

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time in seconds

Bus

4 v

olta

ge in

pu,

imag

inar

y

Bus 4 voltage estimation, imaginary part


(b) Bus 4 voltage real and imaginary parts estimated by three filters in case 1

Figure 4.3: Voltage estimation results of three filters in case 1

112

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time in seconds

MA

E in

pu

Mean Absolute Error (MAE) plots

Traditional KFNaive RKFAKF with InNoVa

(a) MAE comparison of three filters in case 1

(b) MAE histograms of three filters in case 1

Figure 4.4: Voltage estimation accuracy of three filters in case 1

113

Case 2: under-valued initial Q and a malfunctioning device

With the same small Q initialized as above, this time R is set correctly to 5%, expect one

problem: a malfunctioning PMU is installed at bus 4. The bus voltages provided by this PMU

always contain a systematic error, with a normal distribution N (3, 0.12). Fig. 4.5 depicts the

tracking results for bus 4.

In this case, the naive RKF has identified and dropped the bad measurement. However, it

was still inevitably affected by the small Q. Our proposed algorithm not only adjusts Q (the

process noise variance at bus 6 has grown to 0.1126, which is still the largest process noise

variance), but also identifies the bad measurement. Although our algorithm does not exclude

the measurement—because we intend to avoid producing unobservable states, the noise level

of this erroneous measurement becomes 110.18%, which is significantly higher than others

at 5%. Hence this measurement is not trusted and has negligible impact on the state in our

algorithm.

Fig. 4.6 shows that our AKF with InNoVa still offers the best performance. As one may

notice, after the three-phase fault, the naive RKF has a relatively stable MAE level than the

traditional KF, i.e., it has definitely improved the max(MAE) value. Yet still, the under-valued

Q setting has restricted and compromised its performance.

114

12

34

5

−1−0.5

00.5

1

−1

−0.5

0

0.5

1

Time in seconds



Bus

4 v

olta

ge in

pu,

imag

inar

y


(a) Bus 4 voltage estimated by three filter in case 2

0 1 2 3 4 5

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time in seconds

Bus

4 v

olta

ge in

pu,

rea

l



0 1 2 3 4 5

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time in seconds

Bus

4 v

olta

ge in

pu,

imag

inar

y





115

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

1.4

Time in seconds

MA

E in

pu






116

Case 3: fairly-valued initial Q and a malfunctioning device

With the same malfunctioning PMU installed at bus 4, we now have a more appropriate Q

set-up: the process noise variance of each state variable is initialized to be 0.1. Fig. 4.7 displays

the tracking results for bus 4.

We can tell from Fig. 4.7 that when the initial Q is appropriately scaled, the naive RKF

functions properly with respect to bad data detection and correction—it can estimate almost

as accurately as the AKF with InNoVa. From the histogram plot Fig. 4.8(b) we observe more

clearly that the naive RKF performs quite closely to the AKF with InNoVa in this case, while

the traditional KF has suffered from a “shift" caused by the “malfunctioning" PMU data.

117

12

34

5

−1−0.5

00.5

1

−1

−0.5

0

0.5

1

Time in seconds



Bus

4 v

olta

ge in

pu,

imag

inar

y


(a) Bus 4 voltage estimated by three filters in case 3

0 1 2 3 4 5

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time in seconds

Bus

4 v

olta

ge in

pu,

rea

l



0 1 2 3 4 5

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time in seconds

Bus

4 v

olta

ge in

pu,

imag

inar

y





118

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Time in seconds

MA

E in

pu






119

Case 4: over-valued initial Q and a malfunctioning device

Now we increase Q significantly: the process noise variance of each state variable is ini-

tialized to be 1. The malfunctioning PMU behaves the same as in case 2 and 3. The tracking

results for bus 4 are illustrated in Fig. 4.9.

Interestingly in this scenario, the naive RKF almost never detects the systematic error.

Because Q is set so large (hence S is so large) that the normalized innovations always fall

under the threshold. Nevertheless, the systematic error can not escape from the “radar" of

our algorithm: the normalized residual test. In fact, comparing to other measurement noise at

5%, the noise of this erroneous measurement has been inflated to 110.14%. We can tell from

Fig. 4.9 that the AKF with InNoVa still performs the best.

Fig. 4.10(a) and Fig. 4.10(b) further illustrated this benefit: the estimation results produced

by the traditional KF and the naive RKF almost overlap with each other, while the AKF with

InNoVa remains robust and reliable.

120

12

34

5

−1−0.5

00.5

1

−1

−0.5

0

0.5

1

Time in seconds



Bus

4 v

olta

ge in

pu,

imag

inar

y



0 1 2 3 4 5

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time in seconds

Bus

4 v

olta

ge in

pu,

rea

l



0 1 2 3 4 5

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time in seconds

Bus

4 v

olta

ge in

pu,

imag

inar

y





121

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Time in seconds

MA

E in

pu






122

Case 5: over-valued initial Q and complex noise interference

With the same large Q as in case 4, case 5 has more complicated noise interference: beyond

PMU at bus 4, some adjacent PMUs also have bad measurements to affect the estimation. At

t = 1, bus 1 voltage real-part measurements are corrupted by a constant bias value of 5 for 0.1

seconds. At t = 2, bus 4 voltage measurements are corrupted by the normally distributed noise

N (0, 42) for 0.2 seconds. At t = 3, line 4-5 current measurements are corrupted by another

normally distributed noise N (0, 102) for 0.3 seconds. At t = 4, bus 9 voltage imaginary-part

measurements are corrupted by a constant bias value of −3 for 0.4 seconds. Without loss of

generality, the tracking results for bus 4 can be seen in Fig. 4.11.

No doubt the traditional KF is the most vulnerable to this sort of interference. When the

interference is large enough (such as at t = 1, which can be viewed more clearly in the left

figure of Fig. 4.11(b).), the naive RKF is capable of detecting the bad measurements and cal-

ibrating the estimates; otherwise it does not help. Fortunately, the AKF with InNoVa remains

robust to noise interference. Moreover, the inflated R provides useful information for people

to inspect and repair devices, as the measurement noise variance of bus 1, 4, 9 voltages and

line 4-5 current is abnormally larger than others.

Fig. 4.12(a) and Fig. 4.12(b) both verified that the AKF with InNoVa has the most consistent

and highest accuracy level, which leads to its most outstanding performance.

123

12

34

5

−1

0

1

−1

−0.5

0

0.5

1

Time in seconds



Bus

4 v

olta

ge in

pu,

imag

inar

y



0 1 2 3 4 5

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time in seconds

Bus

4 v

olta

ge in

pu,

rea

l



0 1 2 3 4 5

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time in seconds

Bus

4 v

olta

ge in

pu,

imag

inar

y





124

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time in seconds

MA

E in

pu






125

Case 6: failure case

When introducing the typical steps in any state estimation procedure in Section 3.1, we have

mentioned that in contrast to critical measurements, “redundant measurements can be removed

without affecting the system observability". Now it is worth emphasizing that although the

removal of redundant measurements will not affect observability, this does not mean these

measurements are not useful or unnecessary.

In fact, redundant measurements are important in state estimation procedures. Without

enough measurement redundancy, any algorithm will be vulnerable to measurement errors or

telemetry failure. For this reason, in practice state estimation procedures make use of a set of

redundant measurements to filter out such errors and find an optimal estimate.

Although our AKF with InNoVa demonstrates its robustness in many test cases, we should

still be aware that it is not a “silver bullet" if the redundancy of measurements is insufficient.

When multiple interacting bad data are conforming with each other, i.e. multiple interacting

measurements have errors that are coincidentally in agreement, then the AKF with InNoVa

might fail to identify this bad data, and consequently produce incorrect estimation results.

For example in this failure case, we have line 1-4 current measurements (provided by the

PMU at bus 1) which contain a constant error of 5 + 10i, and accidentally, line 4-1 current

measurements (provided by PMU at bus 4) contain a constant error of −5− 10i. Furthermore,

the process noise variance Q is under-valued like in case 1 and 2. Because of inadequate

redundancy of measurements at bus 1, the good bus 1 voltage measurements will be falsely

identified as the bad measurements by the AKF with InNoVa, hence the estimates of bus 1

voltages will largely deviate from the true state, as shown in Fig. 4.13.

Note that since the interacting bad data problem is quite local, it does not affect the per-

formance of the AKF with InNoVa at other buses significantly (see Fig. 4.14), however at bus

126

1 all three algorithms are strongly affected by this particular problem. An obvious solution

would be to increase the redundancy, e.g. to place more PMUs, or to incorporate traditional

RTU measurements in the estimation process, etc.

Similar to that in case 1, Naive RKF has again misjudged some measurements, due to the

overly small Q initialization. As for the AKF with InNoVa, comparing to Fig. 4.4, 4.6, 4.8,

4.10 and 4.12 from previous cases, we can tell its MAE level has been definitely raised, which

is mostly caused by the bad estimating at bus 1.

127

12

34

5

−1

0

1−1

−0.5

0

0.5

1

Time in seconds



Bus

1 v

olta

ge in

pu,

imag

inar

y



0 1 2 3 4 5

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time in seconds

Bus

1 v

olta

ge in

pu,

rea

l



0 1 2 3 4 5

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time in seconds

Bus

1 v

olta

ge in

pu,

imag

inar

y




Figure 4.13: Bus 1 voltage estimation results of three filters in case 6

128

12

34

5

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

Time in seconds



Bus

4 v

olta

ge in

pu,

imag

inar

y



0 1 2 3 4 5

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time in seconds

Bus

4 v

olta

ge in

pu,

rea

l



0 1 2 3 4 5

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time in seconds

Bus

4 v

olta

ge in

pu,

imag

inar

y




Figure 4.14: Bus 4 voltage estimation results of three filters in case 6

129

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

1.4

Time in seconds

MA

E in

pu






130

4.6 My Method: A Two-stage Kalman Filter Approach for Robust andReal-time Power Systems State Tracking

A quasi-steady-state assumption is typically applied to power system operational studies,

for which the state estimation is at the core. Today’s operation is based primarily on a model

that largely ignores dynamics in the power grid—electro-mechanical interaction of genera-

tors and dynamic characteristics of loads and control devices are NOT included in operational

models. This assumption reduces the computation by several orders of magnitude and enables

operation on standard computers to be feasible within the required operational time intervals.

The problem with this assumption is that many studies cannot be performed in the operational

environment. Development of the smart grid makes the grid much less quasi-steady-state, com-

pared with the power grid in the past. In particular, the widespread deployment of renewable

generation, smart load controls, energy storage, and plug-in hybrid vehicles will require funda-

mental changes in the operational concepts and principal components of the grid. Encouraged

by aggressive public policy goals, such as the U.S. State of California’s push to generate 33%

of its energy from renewable sources by 2020, this evolution will continue at an accelerated

speed. This will result in stochastic operating behaviors and dynamics the grid has never seen

nor been designed for.

In this section we developed a two-stage Kalman filter approach to estimate the traditional

states of bus voltage phasor, as well as the dynamic states of generator rotor angle and gener-

ator speed. Theoretically, if we have knowledge of the system noise characteristics, and they

have desirable statistical properties including zero-mean, Gaussian, and spectrally white, then

Kalman filters can achieve its optimal performance. In practice however, the system noise char-

acteristics usually remain unknown to us, and may change from time to time. Thus an adaptive

Kalman filter algorithm becomes favorable in such cases.

In the first stage, we estimate the traditional states from raw PMU measurements, using

131

the Adaptive Kalman Filter with Inflatable Noise Variances (AKF with InNoVa) we have pro-

posed in Section 4.5. As mentioned previously, it is able to identify and reduce the impact of

incorrect system modeling and/or bad PMU measurements. In the next stage, the estimated bus

voltages are fed into an extended Kalman filter to obtain the dynamic state estimation. Simu-

lations demonstrate its robustness to sudden changes of system dynamics and erroneous PMU

measurements.

4.6.1 Stage One

In stage one, the inputs are raw measurement data collected from PMUs installed in the

system. The outputs are the relatively static states: bus voltage magnitudes and phase angles,

which can be estimated using the available measurements and hypothetical system models.

Ideally, the hypothetical models are the “true" models, with accurate noise statistical char-

acteristics. However when the hypothetical models do not match the actual models, and/or

PMU measurements contain significant errors, the estimated states could deviate from the true

states rapidly. Thus we apply the AKF with InNoVa in the first stage to deal with unknown sys-

tem dynamics and erroneous PMU measurements. The detailed implementation is described

in Section 4.5.

4.6.2 Stage Two

In stage two, the estimation results from stage one are fed directly into an Extended Kalman

Filter (EKF) as “measurements". The outputs are the dynamic states: generator rotor angles

and generator speed. As mentioned before, measurements can be expressed in terms of the

state variables either using the rectangular or the polar coordinates. In this method, our mea-

surements are in their rectangular forms, i.e. the real and imaginary parts of all bus voltages,

132

so the process and measurement models are modified as follows.

The Process Model

Without loss of generality, in a power system that consists of n generators, let us consider

the generator i which is connected to the generator terminal bus i. We use a classical model

for the generator composed of a voltage source |Ei|∠δi with constant amplitude behind an

impedance X ′di

. The nonlinear differential-algebraic equations regarding the generator i can be

written asdδidt

= ωB(ωi − ω0)

dωi

dt= ω0

2Hi(Pmi

− |Ei|X′

di

sin δi|Vi| cos θi + |Ei|X′

di

cos δi|Vi| sin θi −Di(ωi − ω0))

= ω0

2Hi(Pmi

− |Ei|X′

di

sin δiRe(Vi) +|Ei|X′

di

cos δiIm(Vi)−Di(ωi − ω0))

(4.24)

where state variables δi and ωi are the generator rotor angle and speed respectively, ωB and

ω0 are the speed base and the synchronous speed in per-unit quantities, Pmiis the mechanical

input, Hi is the machine inertia, Di is the generator damping coefficient and Vi = |Vi|∠θi is

the phasor voltage at the generator terminal bus i (which is a function of δ1, δ2, ..., δn).


change in state can be modeled by the equation

dx

dt= Acx+ wc, (4.25)

where wc is an 2n×1 continuous time process noise vector with 2n×2n noise covariance matrix

Qc = E[wcwTc ], and Ac is an 2n× 2n continuous time state transition Jacobian matrix, whose

133

entries for i ∈ 1, ..., n and j ∈ 1, ..., n (i = j) are the corresponding partial derivatives

Ac[2i−1,2i−1]= 0 (4.26)

Ac[2i−1,2i]= ωB (4.27)

Ac[2i,2i−1]= − ω0|Ei|

2HiX ′di

[cos δiRe(Vi) + sin δi

∂Re(Vi)

∂δi(4.28)

+sin δiIm(Vi)− cos δi∂Im(Vi)

∂δi

]Ac[2i,2i] = −

ω0

2Hi

Di (4.29)

Ac[2i−1,2j−1]= 0 (4.30)

Ac[2i−1,2j]= 0 (4.31)

Ac[2i,2j−1]= − ω0|Ei|

2HiX ′di

[sin δi

∂Re(Vi)

∂δj− cos δi

∂Im(Vi)

∂δj

](4.32)

Ac[2i,2j] = 0. (4.33)

Hence the update of the state vector x from time step (k − 1) to k over duration ∆t has the

complete corresponding discrete-time state transition matrix

A = I + Ac ·∆t. (4.34)

The discrete-time process noise covariance Q can be formulated by integrating the continuous

time process equation (4.25) over the time interval ∆t as described before:

Q =

∫ ∆t

0

eActQceAT

c tdt (4.35)

Hence the process model is written as

xk = Axk−1 + wk−1 (4.36)

= (I + Ac ·∆t)xk−1 + wk−1, (4.37)


134


The expanded system nodal equation can be expressed as:

Yexp

E

V

=

YGG YGL

YLG YLL

E

V

=

IG

0

, (4.38)

where E is the vector of internal generator complex voltages, V is the vector of bus complex

voltages, IG represents electrical currents injected by generators, Yexp is called the expanded

nodal matrix, which includes loads and generator internal impedances. YGG, YGL, YLG and YLL

are the corresponding partitions of the expanded admittance matrix.

According to the expanded system nodal equation, all node voltages phasors can be ex-

pressed in terms of the internal generator voltages and angles by using the bus voltage recon-

struction matrix RV :

V = −Y −1LL YLGE = RVE (4.39)

Thus in a system with n generators and m buses, we have

V1

V2

...

Vm

=

RV11 RV12 · · · RV1n

RV21 RV22 · · · RV2n

...... . . . ...

RVm1 RVm2 · · · RVmn

E1

E2

...

En

, (4.40)

where Vi = |Vi|∠θi is the phasor voltage at bus i, and Ej = |Ej|∠δi is the voltage source at

generator j.

In rectangular form, we have

Re(Vi) = |RVi1||E1| cos(∠RVi1

+ δ1) + |RVi2||E2| cos(∠RVi2

+ δ2) + · · · (4.41)

+ |RVin||En| cos(∠RVin

+ δn)

Im(Vi) = |RVi1||E1| sin(∠RVi1

+ δ1) + |RVi2||E2| sin(∠RVi2

+ δ2) + · · · (4.42)

+ |RVin||En| sin(∠RVin

+ δn)

135

Those “measurements" obtained from stage-one AKF can now be written in this form:

z = h(x) + v (4.43)

where z = [Re(V1), Im(V1), Re(V2), Im(V2), ..., Re(Vm), Im(Vm)]T is the 2m × 1 measure-

ment vector that consists of real and imaginary parts of each estimated bus voltage, x =

[δ1, ω1, δ2, ω2, ..., δn, ωn]T is the state vector and v is the normally distributed measurement

noise with covariance P estimated by stage-one AKF.

After linearization, the measurement model now has this form

z = Hx+ v (4.44)

where H is the corresponding Jacobian matrix:

H =

∂Re(V1)∂δ1

∂Re(V1)∂ω1

· · · ∂Re(V1)∂δn

∂Re(V1)∂ωn

∂Im(V1)∂δ1

∂Im(V1)∂ω1

· · · ∂Im(V1)∂δn

∂Im(V1)∂ωn

∂Re(V2)∂δ1

∂Re(V2)∂ω1

· · · ∂Re(V2)∂δn

∂Re(V2)∂ωn

∂Im(V2)∂δ1

∂Im(V2)∂ω1

· · · ∂Im(V2)∂δn

∂Im(V2)∂ωn

...... . . . ...

...

∂Re(Vm)∂δ1

∂Re(Vm)∂ω1

· · · ∂Re(Vm)∂δn

∂Re(Vm)∂ωn

∂Im(Vm)∂δ1

∂Im(Vm)∂ω1

· · · ∂Im(Vm)∂δn

∂Im(Vm)∂ωn

(4.45)

=

−|RV11 ||E1| sin(∠RV11 + δ1) 0 · · · −|RV1n ||En| sin(∠RV1n + δn) 0

|RV11 ||E1| cos(∠RV11 + δ1) 0 · · · |RV1n ||En| cos(∠RV1n + δn) 0

−|RV21 ||E1| sin(∠RV21 + δ1) 0 · · · −|RV2n ||En| sin(∠RV2n + δn) 0

|RV21 ||E1| cos(∠RV21 + δ1) 0 · · · |RV2n ||En| cos(∠RV2n + δn) 0

...... . . . ...

...

−|RVm1 ||E1| sin(∠RVm1 + δ1) 0 · · · −|RVmn||En| sin(∠RVmn + δn) 0

|RVm1 ||E1| cos(∠RVm1 + δ1) 0 · · · |RVmn||En| cos(∠RVmn + δn) 0

136


Our two-stage Kalman filter approach has been simulated on several multi-machine systems

in various situations. For a more realistic and comprehensive study, this time we will verify the

feasibility of our method on the 16-generator-68-bus New England Test System and New York

Power System model (see Fig. 2.6) from Section 2.8.3, under different abnormal conditions.

Five seconds are simulated in steps of 0.01 seconds. At t = 1.1 seconds a three-phase fault

at bus 29 occurs, and is then cleared at 0.05 seconds. This event represents a large disturbance

(an emergency) causing voltage oscillations and rotor speed changes.

To make it closer to real-world applications, we assume that PMU data contain 1% random

noise. The latest IEEE Standard for Synchrophasor Measurements for Power Systems, IEEE

C37.118.1-2011 [55], has established a criterion of 1% for the value of TVE during calibra-

tion. That means that if the PMU is to be deemed compliant, the observed samples should

not lie outside the circle shown in Fig. 4.2, so that the values calculated by substitution into

Equation (4.22) do not exceed 1%.

In the following five subsections, five abnormal cases are simulated to illustrate the ro-

bustness of our proposed algorithm. In each case we executed three two-stage Kalman filter

approaches for a more informative comparison: Approach 1 uses the traditional KF at stage

one, while Approach 2 uses the naive robust KF (RKF) (using largest normalized innovation

test to identify and exclude bad measurements, without adjusting model parameters) and Ap-

proach 3 uses the AKF with InNoVa.

For each case at stage one, without loss of generality, we visualize the voltage tracking

results for bus 22 in three-dimensional plots. Then the complex voltage MAE and their his-

togram plots are illustrated for a more comprehensive comparison. To maintain consistency of

our experiments in this subsection, the data range of each histogram is set to be [0, 0.15], and

137

the bin (i.e. interval) size is set to be 0.001.

At stage two, the outputs of these three filters are processed by separate EKFs to estimate

the system’s dynamic states. Again without loss of generality, for each case we chose generator

6 to demonstrate and compare the dynamic states tracking results. We have also plotted the

MAE plots and their histograms for generator speed and generator angle estimates separately.

For generator speed, the data range of each histogram is set to be [0, 0.002], and the bin (i.e.

interval) size is set to be 1e−5. For generator angles, the data range of each histograms is set to

be [0, 3.5], and the bin (i.e. interval) size is set to be 0.01.

These test cases are stated in the following subsections respectively.

Case 1: under-valued initial Q and R

We first initialize each state variable with a small variance (2.5e−5), meaning that the user

assumes the system is stable. Then we initialize the PMU measurement noise to 1%, but

simulate actual PMU measurement noise at 2%. In this and the following simulations, we

always use a black solid line for the true state, a red dash-dash line for Approach 1, a green

dash-dot line for Approach 2, and a blue dot-dot line for Approach 3—our proposed approach.

Fig. 4.16 shows the tracking results of the three approaches after stage one. Similar to those

in Fig. 4.3, the small Q has dominated the behavior of the KF and the naive RKF, causing them

to be overconfident with the quasi-static model, where the system states have slow and small

oscillations. Yet by adjusting Q and R, the AKF with InNoVa closely tracks the true state.

We have observed that bus 29 voltages have the largest process variance (0.1180) which is no

surprise because the fault occurs at bus 29. The average measurement noise has grown to 1.4%

in 5 seconds. The accuracy evaluation of stage one estimates can be found in Fig. 4.17.

Fig. 4.18 illustrates the corresponding dynamic state estimates of the three approaches,

138

which are obtained after stage two. Fig. 4.19 has displayed the MAE plots and their histogram

plots respectively, for the generator speed estimates; while Fig. 4.20 has displayed those for the

generator angle estimates. Naturally following the previous stage, Approach 3 still outperforms

the others.

139

12

34

5

0.850.9

0.951

1.05

0.1

0.2

0.3

0.4

0.5

Time in seconds



Bus

22

volta

ge in

pu,

imag

inar

y

True stateApproach 1Approach 2Approach 3

(a) Bus 22 voltage estimated by three approaches after stage one

0 1 2 3 4 50.8

0.85

0.9

0.95

1

1.05

Time in seconds

Bus

22

volta

ge in

pu,

rea

l



0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

Time in seconds

Bus

22

volta

ge in

pu,

imag

inar

y



(b) Voltage real and imaginary parts estimated by three approaches after stage one

Figure 4.16: Voltage estimation results of three approaches in case 1

140

0 1 2 3 4 50

0.02

0.04

0.06

0.08

0.1

0.12

Time in seconds

MA

E in

pu


Approach 1Approach 2Approach 3

(a) MAE comparison of three approaches after stage one

(b) MAE histograms of three approaches after stage one

Figure 4.17: Voltage estimation accuracy of three approaches in case 1

141

0 1 2 3 4 50.998

0.9985

0.999

0.9995

1

1.0005

1.001

1.0015

1.002

Time in seconds

Gen

erat

or s

peed

in p

u

Generator 6 speed estimation


(a) Generator 6 rotor speed estimated by three approaches after stage two

0 1 2 3 4 5

25

30

35

40

45

50

55

Time in seconds

gene

rato

r an

gle

in d

egre

es

Generator 6 angle estimation


(b) Generator 6 rotor angle estimated by three approaches after stage two

Figure 4.18: Rotor speed and angle estimation results of three approaches in case 1

142

0 1 2 3 4 50

0.5

1

1.5x 10

−3

Time in seconds

MA

E in

pu

Mean Absolute Error (MAE) plots, generator speed


(a) Rotor speed MAE comparison of three approaches after stage two

(b) Rotor speed MAE histograms of three approaches after stage two

Figure 4.19: Rotor speed estimation accuracy of three approaches in case 1

143

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

Time in seconds

MA

E in

pu

Mean Absolute Error (MAE) plots, generator angle


(a) Rotor angle MAE comparison of the three approaches after stage two

(b) Rotor angle MAE histograms of the three approaches after stage two

Figure 4.20: Rotor angle estimation results of three approaches in case 1

144

Case 2: under-valued initial Q and a malfunctioning device

While Q is initialized with the same values as above, R is set correctly this time, i.e. PMU

measurements are simulated with 1% noise. We introduce a new problem: a malfunctioning

PMU at bus 22. The simulated voltage measurements provided by this PMU contain systematic

errors, with a normal distribution N (1, 0.12). Fig. 4.21 illustrates the results after stage one.

Fig. 4.21 shows that at stage one the naive RKF has identified and dropped the bad mea-

surements. However it is still affected by the under-valued initial Q. The AKF with In-

NoVa not only adjusted Q (bus 29 has process variance grown to 0.1165, which is still above

the crowd), but also identified the bad measurement. Although our algorithm does not ex-

clude the measurement—we seek to avoid unobservable conditions by maintaining sufficient

redundancy—the noise of this erroneous measurement becomes 41.21%, which is significantly

higher than the others (1%). Hence this measurement is not weighted as heavily, and has a neg-

ligible impact on the estimated states. Fig. 4.22 also confirms that at stage one, while Approach

2 does improve, Approach 3 stays the closest to the truth.

Consequently, the accuracy levels of these three filters have directly affected the estimation

performances at the next stage. The tracking quality remains to be Approach 3 > Approach

2 > Approach 1, as seen in Fig. 4.23, Fig. 4.24 and Fig. 4.25, which show the dynamic states

tracking results, MAE analysis for generator speed estimates and MAE analysis for generator

angle estimates respectively.

145

12

34

5

0.850.9

0.951

1.05

0.1

0.2

0.3

0.4

0.5

Time in seconds



Bus

22

volta

ge in

pu,

imag

inar

y



0 1 2 3 4 50.8

0.85

0.9

0.95

1

1.05

Time in seconds

Bus

22

volta

ge in

pu,

rea

l



0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

Time in seconds

Bus

22

volta

ge in

pu,

imag

inar

y





146

0 1 2 3 4 50

0.02

0.04

0.06

0.08

0.1

0.12

Time in seconds

MA

E in

pu






147

0 1 2 3 4 50.998

0.9985

0.999

0.9995

1

1.0005

1.001

1.0015

1.002

Time in seconds

Gen

erat

or s

peed

in p

u




0 1 2 3 4 5

25

30

35

40

45

50

55

Time in seconds

gene

rato

r an

gle

in d

egre

es





148

0 1 2 3 4 50

0.5

1

1.5x 10

−3

Time in seconds

MA

E in

pu






149

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

Time in seconds

MA

E in

pu






150

Case 3: fairly-valued initial Q and a malfunctioning device

This time the PMU at bus 22 has the same malfunctioning behaviour as described in case 2,

but the initial process noise variance of each state variable is 0.05, which is more appropriate

for our system model.

At stage one, the AKF with InNoVa has its process noise variance at bus 29 grown to

0.1166 in 5 seconds; it also identifies the bad measurements by inflating the corresponding

measurement noise level to 41.17%. As compared to others at 1%, it is quite obvious that we

are having a malfunctioning device at bus 22. Fig. 4.26 tells us that with this fairly-valued

initial Q, while the AKF with InNoVa still performs the best, the naive RKF is able to correctly

detect, identify and eliminate the bad data most of the time. The MAE analysis in Fig.4.27 has

shown that the naive RKF can behave quite close to the AKF with InNoVa in this case, while

the traditional KF produces a shifted tracking result because of the mulfunctioned PMU data.

At stage two, Fig. 4.28 also proves that this Q initialization has greatly helped Approach

2 in its dynamic state estimation. Fig. 4.29 and Fig. 4.30 have provided detailed information

regarding dynamic state estimation accuracy of these three approaches.

151

12

34

5

0.85

0.9

0.95

1

0.1

0.2

0.3

0.4

0.5

Time in seconds



Bus

22

volta

ge in

pu,

imag

inar

y



0 1 2 3 4 50.8

0.85

0.9

0.95

1

1.05

Time in seconds

Bus

22

volta

ge in

pu,

rea

l



0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

Time in seconds

Bus

22

volta

ge in

pu,

imag

inar

y





152

0 1 2 3 4 50

0.005

0.01

0.015

0.02

0.025

Time in seconds

MA

E in

pu






153

0 1 2 3 4 50.998

0.9985

0.999

0.9995

1

1.0005

1.001

1.0015

1.002

Time in seconds

Gen

erat

or s

peed

in p

u




0 1 2 3 4 5

25

30

35

40

45

50

55

Time in seconds

gene

rato

r an

gle

in d

egre

es





154

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−3

Time in seconds

MA

E in

pu






155

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time in seconds

MA

E in

pu






156

Case 4: over-valued initial Q and a malfunctioning device

Now the initial process noise variances in Q are each assigned a much larger value 1, mean-

ing that the user is not confident with the process model. This time the voltage measurement

errors at bus 22 are even larger, with distribution N (2.8, 0.12).

The results obtained from stage one and stage two are depicted in Fig. 4.31 and Fig. 4.33

respectively. Interestingly in Fig. 4.31(a), the naive RKF rarely detects the systematic error.

The reason is that Q is set so large (hence S is so large), the normalized innovations almost al-

ways fall under the threshold. However the error is always detected by the normalized residual

test in the AKF with InNoVa. As a matter of fact, comparing to other measurement noise at

1%, the noise of this erroneous measurement has been inflated to 100.95%.

In MAE analyses plots Fig. 4.32 (for bus voltage estimates), Fig. 4.34 (for generator speed

estimates) and Fig. 4.35 (for generator angle estimates), we can see that in either stage one

or stage two, Approach 3 always performs the best, while Approach 2 performs as bad as

Approach 1 almost all the time.

157

12

34

5

0.850.9

0.951

1.05

0.1

0.2

0.3

0.4

0.5

Time in seconds



Bus

22

volta

ge in

pu,

imag

inar

y



0 1 2 3 4 50.8

0.85

0.9

0.95

1

1.05

Time in seconds

Bus

22

volta

ge in

pu,

rea

l



0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

Time in seconds

Bus

22

volta

ge in

pu,

imag

inar

y





158

0 1 2 3 4 50

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Time in seconds

MA

E in

pu






159

0 1 2 3 4 50.998

0.9985

0.999

0.9995

1

1.0005

1.001

1.0015

1.002

Time in seconds

Gen

erat

or s

peed

in p

u




0 1 2 3 4 5

25

30

35

40

45

50

55

Time in seconds

gene

rato

r an

gle

in d

egre

es





160

0 1 2 3 4 50

0.002

0.004

0.006

0.008

0.01

0.012

Time in seconds

MA

E in

pu






161

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

Time in seconds

MA

E in

pu






162

Case 5: over-valued initial Q and complex noise interference

Although the process noise covariance Q is initialized with the same large values as in

case 4, case 5 has more complicated noise interference (note that although errors are treated as

Gaussian white noise, they do NOT have to be Gaussian and white): at t = 1, bus 22 voltage

real-part measurements are corrupted by a constant bias value of 4 for 0.1 second; at t = 2,

bus 23 voltage measurements are corrupted by the normally distributed noise N (0, 52) for 0.2

second; at t = 3, line 22− 21 current measurements are corrupted by the uniformly distributed

noise U(0, 10) for 0.3 second; at t = 4, line 23 − 22 current imaginary-part measurements

are corrupted by a constant bias value of −15 for 0.4 second. Fig. 4.36 and Fig. 4.38 plot the

tracking results from stage one and stage two respectively.

It is clear that Approach 1 is the most vulnerable to these disturbances. When the distur-

bance is large enough (e.g., t = 1), the naive RKF in Approach 2 is capable of detecting the

bad measurements and calibrating the estimates in stage one, but otherwise it underperforms.

Fortunately, the AKF with InNoVa in Approach 3 remains robust throughout. Moreover, as the

corresponding measurement noise variances suddenly become abnormally large, the inflated R

can be used to signal humans of a need to inspect and repair devices. Fig. 4.37 shows that var-

ious noise disturbances have negligible effect on the AKF with InNoVa. Therefore, Approach

3 still outputs the most impressive dynamic state estimating results with the highest accuracy

level, as seen in Fig. 4.39 (for generator speed estimates) and Fig. 4.35 (for generator angle

estimates).

163

12

34

5

0.850.9

0.951

1.05

0.1

0.2

0.3

0.4

0.5

Time in seconds



Bus

22

volta

ge in

pu,

imag

inar

y



0 1 2 3 4 50.8

0.85

0.9

0.95

1

1.05

Time in seconds

Bus

22

volta

ge in

pu,

rea

l



0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

Time in seconds

Bus

22

volta

ge in

pu,

imag

inar

y





164

0 1 2 3 4 50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Time in seconds

MA

E in

pu






165

0 1 2 3 4 50.998

0.9985

0.999

0.9995

1

1.0005

1.001

1.0015

1.002

Time in seconds

Gen

erat

or s

peed

in p

u




0 1 2 3 4 5

25

30

35

40

45

50

55

Time in seconds

gene

rato

r an

gle

in d

egre

es





166

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2x 10

−3

Time in seconds

MA

E in

pu






167

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

Time in seconds

MA

E in

pu






168

CHAPTER 5

CONCLUSIONS AND FUTURE WORK

In this dissertation I have explored the benefits of on/off-line uncertainty analysis in three

distinct but interrelated research topics, with applications in power systems:

1. Grid Sensor Placement

Chapter 2 presented an approach that combines off-line uncertainty analysis and net-

work topology analysis, to estimate the steady-state performance of candidate sensor

placement design for discrete-space systems. Based on this framework, we restated the

optimal PMU placement problem, applied our method on different system configura-

tions, and visualized the results quantitatively. Taking one step further, we tied it in with

dynamic state estimation to provide example solutions to the industry.

The benefits of our approach include: 1) it can be performed efficiently without even

running the actual estimation procedure; 2) it is generalized for any PMU placement

design, regardless of achieving full observability.

While we have initially chosen to work with PMU measurements only, the future work

could extend the measurement set to include conventional RTU measurements. More-

over, this method is not restricted to PMU placement problems—it can be employed in

different discrete-space system sensor placement optimization frameworks.

2. Filter Computation Adaptation

Chapter 3 presented the Lower Dimensional Measurement-space (LoDiM) state esti-

mation method, a Kalman-filter-based state estimation algorithm with a measurement

selection procedure that can adapt to limited computational resources. Compared to the

169

traditional batch approach of KF/EKF methods, which handles the entire measurement-

space, the LoDiM only needs to deal with a lower-dimensional measurement-space dur-

ing each estimation cycle—the measurement selection procedure guides the LoDiM to

dynamically choose a “most critical" (most effectively reduce the largest estimation un-

certainty component) measurement subset. A smaller measurement-space incorporates

less information each cycle, and hence leads to reduced computation time and increased

reporting rate, which is favorable in various state tracking applications.

Specifically for power system dynamic state estimation problems, where the measurement-

spaces are exceptionally large, we have proposed the Reduced Measurement-space Dy-

namic State Estimation (ReMeDySE) method derived from the LoDiM. Together with

high frequency and accuracy PMU data, the ReMeDySE can facilitate dynamic state

estimation in large-scale power systems.

When some emergency tasks are competing for computational resources with the estima-

tion process, our method allows the estimation to continue with a reduced computational

load, and release some resources for the tasks, without sacrificing the reporting rate.

Then when the tasks are completed, the estimation process can reclaim the resources,

so that the filter computation can go back up as desired. We can optimally adapt our

algorithm to the available computation budget by analyzing the trade-offs.

Although the LoDiM approach is presented in the context of the Kalman filter, the as-

sociated measurement selection procedure is not filter-specific, i.e. it can be used with

other state estimation methods such as particle and unscented filters. Selecting certain

subspaces inherently delays the selection of the entire (remaining) measurement-space,

which could potentially delay the observance of an important state change. One possi-

ble solution is to investigate the addition of measurement prioritization based on factors

such as unexpected (deviating from prediction) state changes throughout the entire state

space. For example, one can adjust the uncertainty parameters as discussed in our next

topic, to affect priorities for parts of the state space, causing measurement selection to

170

prioritize certain areas.

With all the matrix arithmetic operations, it is possible to further parallelize and opti-

mize the algorithm by taking advantage of modern computational hardware such as the

GPUs. Moreover, we can combine our approach with hierarchical/distributed estimation

methods, making it particularly attractive in large-scale real-time state estimation, for

wide-area power systems and beyond.

3. Adaptive and Robust Estimation

Chapter 4 presented a novel Adaptive Kalman Filter with Inflatable Noise Variances

(AKF with InNoVa). With real-time phasor measurements, this algorithm enables conve-

nient and on-line robust power system state estimation under various adverse conditions,

given sufficient redundancy among measurements.

The AKF with InNoVa is designed to identify and reduce the impact of incorrect system

modeling, as well as erroneous measurements. It is capable of doing so because besides

the regular normalized innovation test, we also adopt a normalized residual test to help

separating the process and measurement factors. With these tests, our algorithm effi-

ciently adjusts the process and measurement noise parameters on-the-fly separately for

more robust estimates. Specifically, the inflation of process noise covariance Q indicates

fast changing state or even wrong model, making it practical for people to localize system

anomalies, e.g. faults, load changes, electricity theft, etc; while the inflation of measure-

ment noise covariance R implies potentially bad measurements. At the same time, an

exponential decay process is employed to enable automatic deflation of the parameters

if the modeling/devices problems are resolved.

To produce real-time and robust estimation of more dynamic states besides traditional

static states, we have combined the AKF with InNoVa with power system dynamic state

estimation from our previous work. The result is a more comprehensive power system

state estimation technique with a two-stage Kalman filter approach. The AKF with In-

171

NoVa in stage one deals with system modeling and measurement errors. It gives more

accurate estimates on bus voltage phasors, which are also served as the input to the

EKF in stage two, to generate more accurate estimates on generator rotor speed and an-

gles. The output of both stages are essential for myriad time-critical applications. For

example in contingency analysis, signatures in different types of contingency may be

detected from the change in state estimates. In fact, nowadays in the age of social media,

besides the “official" measurements from the industrial control systems, public informa-

tion available on the internet is a new source of measurements that can help to confirm

the contingency and hence alert the operators.

The AKF with InNoVa should not be limited to power system state estimation. Fur-

thermore, with the advance of computational hardware technologies and optimization

algorithms, our approach is subject to further modifications.

172

REFERENCES

[1] Greg Welch, Gary Bishop, Leandra Vicci, Stephen Brumback, Kurtis Keller, and D’nardoColucci. High-performance wide-area optical tracking: The hiball tracking system.Teleoperators and Virtual Environments, 10(1), February 2001.

[2] A.G. Phadke, J.S. Thorp, and M.G. Adamiak. A new measurement technique for track-ing voltage phasors, local system frequency, and rate of change of frequency. IEEETransactions on Power Apparatus and Systems, PAS-102(5), May 1983.

[3] A. G. Phadke, J. S. Thorp, and K. J. Karimi. State estimlatjon with phasor measurements.IEEE Transactions on Power Systems, 1(1), February 1986.

[4] T.L. Baldwin, L. Mili, M.B. Boisen Jr., and R. Adapa. Power system observability withminimal phasor measurement placement. IEEE Transactions on Power Systems, 8(2),May 1993.

[5] Jian Chen and Ali Abur. Placement of pmus to enable bad data detection in state estima-tion. IEEE Transactions on Power Systems, 21(4), November 2006.

[6] Bei Gou. Generalized integer linear programming formulation for optimal pmu place-ment. IEEE Transactions on Power Systems, 23(3), August 2008.

[7] Bei Xu and Ali Abur. Observability analysis and measurement placement for systemswith pmus. In Power Systems Conference and Exposition, 2004. IEEE PES, October2004.

[8] Jinghe Zhang, Greg Welch, and Gary Bishop. Observability and estimation uncertaintyanalysis for pmu placement alternatives. 2010 North American Power Symposium(NAPS 2010), Arlington, TX, USA, September 2010.

[9] Bonnie Danette Allen and Greg Welch. A general method for comparing the expectedperformance of tracking and motion capture systems. VRST ’05: the ACM symposiumon Virtual reality software and technology, Monterey, CA, USA, 2005.

[10] Bonnie Danette Allen. Hardware Design Optimization for Human Motion Tracking Sys-tems. Ph.d. dissertation, The University of North Carolina at Chapel Hill, Departmentof Computer Science, Chapel Hill, NC, USA, November 2007.

[11] Zhenyu Huang, Kevin Schneider, and Jarek Nieplocha. Feasibility studies of apply-ing kalman filter techniques to power system dynamic state estimation. InternationalPower Engineering Conference 2007 (IPEC 2007), December 2007.

[12] Jinghe Zhang, Greg Welch, Gary Bishop, and Zhenyu Huang. Optimal pmu placementevaluation for power system dynamic state estimation. IEEE PES Conference on In-novative Smart Grid Technologies Europe (ISGT 2010), Goteborg, Sweden, October2010.

173

[13] Rudolph Emil Kalman. A new approach to linear filtering and prediction problems. Trans-action of the ASME—Journal of Basic Engineering, Series D, 82, 1960.

[14] Gregory Francis Welch. SCAAT: Incremental Tracking with Incomplete Information.Ph.d. dissertation, The University of North Carolina at Chapel Hill, Department ofComputer Science, Chapel Hill, NC, USA, October 1996.

[15] Jinghe Zhang, Greg Welch, and Gary Bishop. Power system state estimation with dy-namic optimal measurement selection. 2011 IEEE Symposium on Computational In-telligence Applications in Smart Grid, Paris, France, April 2011.

[16] Jinghe Zhang, Greg Welch, and Gary Bishop. Lodim: A novel power system state es-timation method with dynamic measurement selection. 2011 IEEE Power & EnergySociety General Meeting, Detroit, MI, USA, July 2011.

[17] Jinghe Zhang, Greg Welch, Gary Bishop, and Zhenyu Huang. Reduced measurement-space dynamic state estimation (remedyse) for power systems. IEEE PowerTech 2011,Trondheim, Norway, June 2011.

[18] Chakphed Madtharad, Suttichai Premrudeepreechacharn, and Neville R. Watson. Powersystem state estimation using singular value decomposition. Electric Power SystemsResearch, 67(2), 2003.

[19] Greg Welch and Gary Bishop. An introduction to the kalman filter. Technical report,University of North Carolina at Chapel Hill, Department of Computer Science, 1995.

[20] Mohinder S. Grewal and Angus P. Andrews. Kalman Filtering Theory and Practice.Information and System Sciences Series. Prentice Hall, Upper Saddle River, NJ USA,1993.

[21] Ali Abur and Antonio Gomez Exposito. Power System State Estimation: Theory andImplementation. Marcel Dekker, 2004.

[22] Jin Xu, Miles H. F. Wen, Victor O. K. Li, and Ka-Cheong Leung. Optimal pmu placementfor wide-area monitoring using chemical reaction optimization. Innovative Smart GridTechnologies (ISGT), 2013 IEEE PES, February 2013.

[23] Ashwani Kumara, Biswarup Dasb, and Jaydev Sharmab. Dynamic state estimation ofpower system harmonics with bad data. Electric Power Components and Systems,33(12), December 2005.

[24] Graham Rogers. Power System Oscillation. Academic Publishers, Boston, 2000.

[25] PNNL. Kalman equations. Personal Communications with the Electricity Infrastructuregroup, Pacific Northwest National Laboratory, 2008.

[26] Barbara Chapman, Gabriele Jost, and Ruud van der Pas. Using OpenMP: Portable SharedMemory Parallel Programming. the MIT Press, 2007.

174

[27] Jarek Nieplocha, Bruce Palmer, Manojkumar Krishnan, and P. Saddayappan. Overviewof the global arrays parallel aoftware development toolkit. 2006 ACM/IEEE conferenceon Supercomputing SC ’06, Tampa, Florida, 2006.

[28] Anant Oonsivilai and Kenedy A. Greyson. Optimization of distributed processors forpower system: Kalman filters using petri net. World Academy of Science, Engineeringand Technology 53, 2009.

[29] M.J. Goris, D.A. Gray, and I.M.Y. Mareels. Reducing the computational load of a kalmanfilter. Electronics Letters, 33(18), August 1997.

[30] Shoujun Bian and Michael A. Henson. Measurement selection for on-line estimationof nonlinear wave models for high purity distillation columns. Chemical EngineeringScience, 61(10), 2006.

[31] Ning Zhou, Zhenyu Huang, Greg Welch, and Jinghe Zhang. Identifying optimal mea-surement subspace for ensemble kalman filter. Electronics Letters, 48(11), May 2012.

[32] Atif S. Debs and Robert E. Larson. A dynamic estimation for tracking the state of apower system. IEEE Transactions on Power Apparatus and Systems, vol. PAS-89, no.7, September/October 1970.

[33] Amit Jain and Shivakumar N. R. Power system tracking and dynamic state estimation. InPower Systems Conference and Exposition (PSCE ’09). IEEE/PES, March 2009.

[34] Zhenyu Huang, Pengwei Du, Dmitry Kosterev, and Bo Yang. Application of extendedkalman filter techniques for dynamic model parameter calibration. IEEE Power andEnergy Society General Meeting 2009, Calgary, Alberta, Canada, July 2009.

[35] Greg Welch and Gary Bishop. An Introduction to the Kalman Filter. SIGGRAPH 2001course 8. In Computer Graphics, Annual Conference on Computer Graphics & Inter-active Techniques. ACM Press, Addison-Wesley, Los Angeles, CA, USA, 2001.

[36] Frank L. Lewis. Optimal Estimation: With an Introduction to Stochastic Control Theory.John Wiley and Sons, Inc., 1986.

[37] Gary Bishop. The Self-Tracker: A Smart Optical Sensor on Silicon. Ph.d. dissertation,The University of North Carolina at Chapel Hill, Department of Computer Science,Chapel Hill, NC, USA, 1984.

[38] Ricardo Todling. Estimation theory and foundations of atmospheric data assimilation.Technical report, Data Assimilation Office, Goddard Laboratory for Atmospheres,1999.

[39] Mohinder S. Grewal and Augus P. Andrews. Kalman Filtering: Theory and PracticeUsing MATLAB. John Wiley & Sons, Inc., Hoboken, New Jersey, 2008.

[40] Gene H. Golub and Charles F. Van Loan. Matrix Computations. Johns Hopkins Univ.Press, Baltimore, MD, 1989.

175

[41] Saman A. Zonouz and William H. Sanders. A kalman-based coordination for hierarchicalstate estimation: Algorithm and analysis. The 41st Hawaii International Conferenceon System Sciences, 2008.

[42] S. Lakshminarasimhan and A. A. Girgis. Hierarchical state estimation applied to wide-area power systems. IEEE Power Engineering Society General Meeting, June 2007.

[43] M. S. Kurzyn. Real-time state estimation for large power systems using multicomputersystems. Computers & Electrical Engineering, 8(4), 1981.

[44] M. S. Kurzyn. Real-time state estimation for large-scale power systems. IEEE Transac-tions on Power Apparatus and Systems, July 1983.

[45] Jinghe Zhang, Greg Welch, Gary Bishop, and Zhenyu Huang. A two-stage kalman filterapproach for robust and real-time power system state estimation. IEEE Transactionson Sustainable Energy, 2013.

[46] G. E. P. Box and N. R. Draper. Empirical Model Building and Response Surfaces. JohnWiley & Sons, New York, NY, 1987.

[47] Zhongzhi Li and Xuegang Wang. Reverse prediction adaptive kalman filtering algorithmfor maneuvering target tracking. Journal of Computational Information Systems 6:10(2010) 3257-3265., 2010.

[48] Kuang-Rong Shih and Shyh-Jier Huang. Application of a robust algorithm for dynamicstate estimation of a power system. IEEE Power Engineering Review , vol. 22, no. 1,2002.

[49] M. Oussalah and J. De Schutter. Adaptive kalman filter for noise identification. Interna-tional Conference on Noise and Vibration Engineering, September 2000.

[50] Ashwani Kumar, Biswarup Das, and Jaydev Sharma. Robust dynamic state estimationof power system harmonics. International Journal of Electrical Power and EnergySystems Volume 28, Issue 1, January 2006.

[51] Jun Zhu and Ali Abur. Bad data identification when using phasor measurements. PowerTech, 2007 IEEE Lausanne, July 2007.

[52] Hui Xue, Qing quan Jia, Ning Wang, Zhi qian Bo, Hai tang Wang, and Hong xia Ma. Adynamic state estimation method with pmu and scada measurement for power systems.the 8th International Power Engineering Conference (IPEC 2007), Singapore, 2007.

[53] Amit Jain and Shivakumar N. R. Impact of pmus in dynamic state estimation of powersystems. North American Power Symposium (NAPS), Calgary, Canada, September2008.

[54] Antonio Gomez-Exposito, Ali Abur, Patricia Rousseaux, Antonio de la Villa Jaen, andCatalina Gomez-Quiles. On the use of pmus in power system state estimation. 17thPower Systems Computation Conference, Stockholm, Sweden, August 2011.

176

[55] Power System Relaying Committee. C37.118.1-2011: Ieee standard for synchrophasormeasurements for power systems. IEEE Standard Association, December 2011.

[56] J. Peppanen, T. Alquthami, D. Molina, and R. Harley. Optimal pmu placement withbinary pso. IEEE Energy Conversion Congress and Exposition (ECCE), 2012.

[57] F. Aminifar, M. Fotuhi-Firuzabad, and A. Safdarian. Optimal pmu placement based onprobabilistic cost/benefit analysis. IEEE Transactions on Power Systems, February2013.

[58] Paul D. Groves. Principles of GNSS, Inertial, and Multisensor Integrated NavigationSystems. Artech House, 2013.

177

Documents

UNCERTAINTY-DRIVEN ADAPTIVE ESTIMATION WITH …